Integrand size = 34, antiderivative size = 548 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {\left (4 a d^2 (C d-3 c D)+3 b \left (6 c^2 C d-3 B c d^2+A d^3-10 c^3 D\right )\right ) \sqrt {a+b x^2}}{3 d^6}+\frac {\left (5 a d^2 D-4 b \left (3 c C d-B d^2-6 c^2 D\right )\right ) x \sqrt {a+b x^2}}{8 d^5}+\frac {b (C d-3 c D) x^2 \sqrt {a+b x^2}}{3 d^4}+\frac {b D x^3 \sqrt {a+b x^2}}{4 d^3}-\frac {\left (b c^2+a d^2\right ) \left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \sqrt {a+b x^2}}{2 d^6 (c+d x)^2}+\frac {\left (2 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (9 c^2 C d-7 B c d^2+5 A d^3-11 c^3 D\right )\right ) \sqrt {a+b x^2}}{2 d^6 (c+d x)}+\frac {\left (3 a^2 d^4 D-12 a b d^2 \left (3 c C d-B d^2-6 c^2 D\right )-8 b^2 c \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{8 \sqrt {b} d^7}-\frac {\left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (19 c^2 C d-9 B c d^2+3 A d^3-33 c^3 D\right )+2 b^2 c^2 \left (10 c^2 C d-6 B c d^2+3 A d^3-15 c^3 D\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^7 \sqrt {b c^2+a d^2}} \] Output:
1/3*(4*a*d^2*(C*d-3*D*c)+3*b*(A*d^3-3*B*c*d^2+6*C*c^2*d-10*D*c^3))*(b*x^2+ a)^(1/2)/d^6+1/8*(5*a*d^2*D-4*b*(-B*d^2+3*C*c*d-6*D*c^2))*x*(b*x^2+a)^(1/2 )/d^5+1/3*b*(C*d-3*D*c)*x^2*(b*x^2+a)^(1/2)/d^4+1/4*b*D*x^3*(b*x^2+a)^(1/2 )/d^3-1/2*(a*d^2+b*c^2)*(A*d^3-B*c*d^2+C*c^2*d-D*c^3)*(b*x^2+a)^(1/2)/d^6/ (d*x+c)^2+1/2*(2*a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)+b*c*(5*A*d^3-7*B*c*d^2+9*C *c^2*d-11*D*c^3))*(b*x^2+a)^(1/2)/d^6/(d*x+c)+1/8*(3*a^2*d^4*D-12*a*b*d^2* (-B*d^2+3*C*c*d-6*D*c^2)-8*b^2*c*(3*A*d^3-6*B*c*d^2+10*C*c^2*d-15*D*c^3))* arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/d^7-1/2*(2*a^2*d^4*(C*d-3*D*c)+ a*b*d^2*(3*A*d^3-9*B*c*d^2+19*C*c^2*d-33*D*c^3)+2*b^2*c^2*(3*A*d^3-6*B*c*d ^2+10*C*c^2*d-15*D*c^3))*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a )^(1/2))/d^7/(a*d^2+b*c^2)^(1/2)
Time = 5.25 (sec) , antiderivative size = 472, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 \left (-156 c^3 D+c^2 d (68 C-249 D x)-2 c d^2 (6 B+x (-56 C+33 D x))+d^3 \left (-12 A+x \left (-24 B+32 C x+15 D x^2\right )\right )\right )-2 b \left (180 c^5 D-30 c^4 d (4 C-9 D x)+12 c^3 d^2 (6 B+5 x (-3 C+D x))-d^5 x^2 (12 A+x (6 B+x (4 C+3 D x)))+2 c d^4 x (-27 A+x (12 B+x (5 C+3 D x)))-c^2 d^3 (36 A+x (-108 B+5 x (8 C+3 D x)))\right )\right )}{(c+d x)^2}+\frac {24 \left (-2 a^2 d^4 (C d-3 c D)+2 b^2 c^2 \left (-10 c^2 C d+6 B c d^2-3 A d^3+15 c^3 D\right )+a b d^2 \left (-19 c^2 C d+9 B c d^2-3 A d^3+33 c^3 D\right )\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}-\frac {3 \left (3 a^2 d^4 D+12 a b d^2 \left (-3 c C d+B d^2+6 c^2 D\right )+8 b^2 c \left (-10 c^2 C d+6 B c d^2-3 A d^3+15 c^3 D\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{\sqrt {b}}}{24 d^7} \] Input:
Integrate[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^3,x]
Output:
((d*Sqrt[a + b*x^2]*(a*d^2*(-156*c^3*D + c^2*d*(68*C - 249*D*x) - 2*c*d^2* (6*B + x*(-56*C + 33*D*x)) + d^3*(-12*A + x*(-24*B + 32*C*x + 15*D*x^2))) - 2*b*(180*c^5*D - 30*c^4*d*(4*C - 9*D*x) + 12*c^3*d^2*(6*B + 5*x*(-3*C + D*x)) - d^5*x^2*(12*A + x*(6*B + x*(4*C + 3*D*x))) + 2*c*d^4*x*(-27*A + x* (12*B + x*(5*C + 3*D*x))) - c^2*d^3*(36*A + x*(-108*B + 5*x*(8*C + 3*D*x)) ))))/(c + d*x)^2 + (24*(-2*a^2*d^4*(C*d - 3*c*D) + 2*b^2*c^2*(-10*c^2*C*d + 6*B*c*d^2 - 3*A*d^3 + 15*c^3*D) + a*b*d^2*(-19*c^2*C*d + 9*B*c*d^2 - 3*A *d^3 + 33*c^3*D))*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b* c^2) - a*d^2]])/Sqrt[-(b*c^2) - a*d^2] - (3*(3*a^2*d^4*D + 12*a*b*d^2*(-3* c*C*d + B*d^2 + 6*c^2*D) + 8*b^2*c*(-10*c^2*C*d + 6*B*c*d^2 - 3*A*d^3 + 15 *c^3*D))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/Sqrt[b])/(24*d^7)
Time = 1.70 (sec) , antiderivative size = 819, normalized size of antiderivative = 1.49, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2182, 25, 2182, 25, 27, 682, 27, 682, 27, 719, 224, 219, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{3/2} \left (2 \left (\frac {b c^2}{d}+a d\right ) D x^2+\left (a (2 C d-2 c D)-b \left (\frac {5 D c^3}{d^2}-\frac {5 C c^2}{d}+3 B c-3 A d\right )\right ) x+2 \left (A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^2}dx}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{3/2} \left (2 \left (\frac {b c^2}{d}+a d\right ) D x^2+\left (2 a (C d-c D)-b \left (\frac {5 D c^3}{d^2}-\frac {5 C c^2}{d}+3 B c-3 A d\right )\right ) x+2 \left (A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^2}dx}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle \frac {\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}-\frac {\int -\frac {\left (\left (2 d (C d-2 c D) a^2+\frac {b c \left (-5 D c^2+3 C d c-B d^2\right ) a}{d}+A b \left (2 b c^2+3 a d^2\right )\right ) d^2+2 \left (a^2 D d^4-2 a b \left (-7 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+2 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}dx}{a d^2+b c^2}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (d \left (2 a^2 (C d-2 c D) d^2+A b \left (2 b c^2+3 a d^2\right ) d+a b c \left (-5 D c^2+3 C d c-B d^2\right )\right )+2 \left (a^2 D d^4-2 a b \left (-7 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+2 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{d^2 (c+d x)}dx}{a d^2+b c^2}+\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (d \left (2 a^2 (C d-2 c D) d^2+A b \left (2 b c^2+3 a d^2\right ) d+a b c \left (-5 D c^2+3 C d c-B d^2\right )\right )+2 \left (a^2 D d^4-2 a b \left (-7 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+2 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{c+d x}dx}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {2 b \left (b c^2+a d^2\right ) \left (a d \left (a (4 C d-9 c D) d^2+b \left (-15 D c^3+10 C d c^2-6 B d^2 c+6 A d^3\right )\right )+\left (3 a^2 D d^4-a b \left (-51 D c^2+28 C d c-12 B d^2\right ) d^2-4 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{4 b d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a^2 d^4 D-2 a b d^2 \left (-2 B d^2-7 c^2 D+4 c C d\right )-b^2 c \left (2 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+2 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{6 d^2}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\left (a d^2+b c^2\right ) \int \frac {\left (a d \left (a (4 C d-9 c D) d^2+b \left (-15 D c^3+10 C d c^2-6 B d^2 c+6 A d^3\right )\right )+\left (3 a^2 D d^4-a b \left (-51 D c^2+28 C d c-12 B d^2\right ) d^2-4 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a^2 d^4 D-2 a b d^2 \left (-2 B d^2-7 c^2 D+4 c C d\right )-b^2 c \left (2 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+2 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{6 d^2}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\frac {\left (a d^2+b c^2\right ) \left (\frac {\int \frac {b \left (b c^2+a d^2\right ) \left (a d \left (a (8 C d-21 c D) d^2+4 b \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right )+\left (3 a^2 D d^4-12 a b \left (-6 D c^2+3 C d c-B d^2\right ) d^2-8 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (3 a^2 d^4 D-a b d^2 \left (-12 B d^2-51 c^2 D+28 c C d\right )-4 b^2 c \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+4 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{2 d^2}\right )}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a^2 d^4 D-2 a b d^2 \left (-2 B d^2-7 c^2 D+4 c C d\right )-b^2 c \left (2 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+2 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{6 d^2}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\left (a d^2+b c^2\right ) \left (\frac {\left (a d^2+b c^2\right ) \int \frac {a d \left (a (8 C d-21 c D) d^2+4 b \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right )+\left (3 a^2 D d^4-12 a b \left (-6 D c^2+3 C d c-B d^2\right ) d^2-8 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (3 a^2 d^4 D-a b d^2 \left (-12 B d^2-51 c^2 D+28 c C d\right )-4 b^2 c \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+4 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{2 d^2}\right )}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a^2 d^4 D-2 a b d^2 \left (-2 B d^2-7 c^2 D+4 c C d\right )-b^2 c \left (2 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+2 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{6 d^2}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {\frac {\frac {\left (a d^2+b c^2\right ) \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {\left (3 a^2 d^4 D-12 a b d^2 \left (-B d^2-6 c^2 D+3 c C d\right )-8 b^2 c \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}+\frac {4 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}\right )}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (3 a^2 d^4 D-a b d^2 \left (-12 B d^2-51 c^2 D+28 c C d\right )-4 b^2 c \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+4 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{2 d^2}\right )}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a^2 d^4 D-2 a b d^2 \left (-2 B d^2-7 c^2 D+4 c C d\right )-b^2 c \left (2 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+2 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{6 d^2}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\left (2 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-7 D c^3+5 C d c^2-3 B d^2 c+A d^3\right )\right ) \left (b x^2+a\right )^{5/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}+\frac {\frac {\left (2 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right )+3 d \left (a^2 D d^4-2 a b \left (-7 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+2 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{6 d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\sqrt {b x^2+a} \left (4 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right )+d \left (3 a^2 D d^4-a b \left (-51 D c^2+28 C d c-12 B d^2\right ) d^2-4 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) x\right )}{2 d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {4 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {\left (3 a^2 D d^4-12 a b \left (-6 D c^2+3 C d c-B d^2\right ) d^2-8 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{2 d^2}\right )}{2 d^2}}{d^2 \left (b c^2+a d^2\right )}}{2 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 \left (b c^2+a d^2\right ) (c+d x)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {\left (a d^2+b c^2\right ) \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {4 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a^2 d^4 D-12 a b d^2 \left (-B d^2-6 c^2 D+3 c C d\right )-8 b^2 c \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{\sqrt {b} d}\right )}{2 d^2}+\frac {\sqrt {a+b x^2} \left (d x \left (3 a^2 d^4 D-a b d^2 \left (-12 B d^2-51 c^2 D+28 c C d\right )-4 b^2 c \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+4 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{2 d^2}\right )}{2 d^2}+\frac {\left (a+b x^2\right )^{3/2} \left (3 d x \left (a^2 d^4 D-2 a b d^2 \left (-2 B d^2-7 c^2 D+4 c C d\right )-b^2 c \left (2 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )+2 \left (2 a^2 d^4 (C d-3 c D)+a b d^2 \left (3 A d^3-9 B c d^2-33 c^3 D+19 c^2 C d\right )+2 b^2 c^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )\right )}{6 d^2}}{d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{5/2} \left (2 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (A d^3-3 B c d^2-7 c^3 D+5 c^2 C d\right )\right )}{d^2 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{2 d^2 (c+d x)^2 \left (a d^2+b c^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\frac {\left (2 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-7 D c^3+5 C d c^2-3 B d^2 c+A d^3\right )\right ) \left (b x^2+a\right )^{5/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}+\frac {\frac {\left (2 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right )+3 d \left (a^2 D d^4-2 a b \left (-7 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+2 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{6 d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\sqrt {b x^2+a} \left (4 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right )+d \left (3 a^2 D d^4-a b \left (-51 D c^2+28 C d c-12 B d^2\right ) d^2-4 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) x\right )}{2 d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (3 a^2 D d^4-12 a b \left (-6 D c^2+3 C d c-B d^2\right ) d^2-8 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} d}-\frac {4 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}\right )}{2 d^2}\right )}{2 d^2}}{d^2 \left (b c^2+a d^2\right )}}{2 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 \left (b c^2+a d^2\right ) (c+d x)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (2 a \left (-3 D c^2+2 C d c-B d^2\right ) d^2+b c \left (-7 D c^3+5 C d c^2-3 B d^2 c+A d^3\right )\right ) \left (b x^2+a\right )^{5/2}}{d^2 \left (b c^2+a d^2\right ) (c+d x)}+\frac {\frac {\left (2 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right )+3 d \left (a^2 D d^4-2 a b \left (-7 D c^2+4 C d c-2 B d^2\right ) d^2-b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+2 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}}{6 d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\sqrt {b x^2+a} \left (4 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right )+d \left (3 a^2 D d^4-a b \left (-51 D c^2+28 C d c-12 B d^2\right ) d^2-4 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) x\right )}{2 d^2}+\frac {\left (b c^2+a d^2\right ) \left (\frac {\left (3 a^2 D d^4-12 a b \left (-6 D c^2+3 C d c-B d^2\right ) d^2-8 b^2 c \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right )}{\sqrt {b} d}-\frac {4 \left (2 a^2 (C d-3 c D) d^4+a b \left (-33 D c^3+19 C d c^2-9 B d^2 c+3 A d^3\right ) d^2+2 b^2 c^2 \left (-15 D c^3+10 C d c^2-6 B d^2 c+3 A d^3\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right )}{d \sqrt {b c^2+a d^2}}\right )}{2 d^2}\right )}{2 d^2}}{d^2 \left (b c^2+a d^2\right )}}{2 \left (b c^2+a d^2\right )}-\frac {\left (-D c^3+C d c^2-B d^2 c+A d^3\right ) \left (b x^2+a\right )^{5/2}}{2 d^2 \left (b c^2+a d^2\right ) (c+d x)^2}\) |
Input:
Int[((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^3,x]
Output:
-1/2*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a + b*x^2)^(5/2))/(d^2*(b*c^2 + a*d^2)*(c + d*x)^2) + (((2*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(5*c^2 *C*d - 3*B*c*d^2 + A*d^3 - 7*c^3*D))*(a + b*x^2)^(5/2))/(d^2*(b*c^2 + a*d^ 2)*(c + d*x)) + (((2*(2*a^2*d^4*(C*d - 3*c*D) + a*b*d^2*(19*c^2*C*d - 9*B* c*d^2 + 3*A*d^3 - 33*c^3*D) + 2*b^2*c^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D)) + 3*d*(a^2*d^4*D - 2*a*b*d^2*(4*c*C*d - 2*B*d^2 - 7*c^2*D) - b^2*c*(10*c^2*C*d - 6*B*c*d^2 + 2*A*d^3 - 15*c^3*D))*x)*(a + b*x^2)^(3/2)) /(6*d^2) + ((b*c^2 + a*d^2)*(((4*(2*a^2*d^4*(C*d - 3*c*D) + a*b*d^2*(19*c^ 2*C*d - 9*B*c*d^2 + 3*A*d^3 - 33*c^3*D) + 2*b^2*c^2*(10*c^2*C*d - 6*B*c*d^ 2 + 3*A*d^3 - 15*c^3*D)) + d*(3*a^2*d^4*D - a*b*d^2*(28*c*C*d - 12*B*d^2 - 51*c^2*D) - 4*b^2*c*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*x)*Sqr t[a + b*x^2])/(2*d^2) + ((b*c^2 + a*d^2)*(((3*a^2*d^4*D - 12*a*b*d^2*(3*c* C*d - B*d^2 - 6*c^2*D) - 8*b^2*c*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^ 3*D))*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) - (4*(2*a^2*d^4*(C *d - 3*c*D) + a*b*d^2*(19*c^2*C*d - 9*B*c*d^2 + 3*A*d^3 - 33*c^3*D) + 2*b^ 2*c^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 - 15*c^3*D))*ArcTanh[(a*d - b*c*x) /(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d*Sqrt[b*c^2 + a*d^2])))/(2*d^2) ))/(2*d^2))/(d^2*(b*c^2 + a*d^2)))/(2*(b*c^2 + a*d^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2943\) vs. \(2(508)=1016\).
Time = 1.49 (sec) , antiderivative size = 2944, normalized size of antiderivative = 5.37
Input:
int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
D/d^3*(1/4*x*(b*x^2+a)^(3/2)+3/4*a*(1/2*x*(b*x^2+a)^(1/2)+1/2*a/b^(1/2)*ln (b^(1/2)*x+(b*x^2+a)^(1/2))))+1/d^4*(C*d-3*D*c)*(1/3*(b*(x+c/d)^2-2*b*c/d* (x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)-b*c/d*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+ c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2 -4*b^2*c^2/d^2)/b^(3/2)*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d *(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)))+(a*d^2+b*c^2)/d^2*((b*(x+c/d)^2-2*b*c/ d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/ 2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-(a*d^2+b*c^2)/d^ 2/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a* d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2 ))/(x+c/d))))+1/d^5*(B*d^2-2*C*c*d+3*D*c^2)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)* (b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(5/2)-3*b*c*d/(a*d^2+b*c^2 )*(1/3*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)-b*c/d*(1/4*(2 *b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2 )+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln((-b*c/d+b*(x+c/d))/ b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)))+(a*d^2+b*c ^2)/d^2*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b^(1/2)*c/d *ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/ d^2)^(1/2))-(a*d^2+b*c^2)/d^2/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2 )/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\text {Timed out} \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{3}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**3,x)
Output:
Integral((a + b*x**2)**(3/2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 1889 vs. \(2 (510) = 1020\).
Time = 0.18 (sec) , antiderivative size = 1889, normalized size of antiderivative = 3.45 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="maxi ma")
Output:
-3/2*sqrt(b*x^2 + a)*D*b^2*c^5/(b*c^2*d^6 + a*d^8) + 3/2*sqrt(b*x^2 + a)*D *b^2*c^4*x/(b*c^2*d^5 + a*d^7) - 1/2*(b*x^2 + a)^(3/2)*D*b*c^4/(b*c^2*d^5* x + a*d^7*x + b*c^3*d^4 + a*c*d^6) + 3/2*sqrt(b*x^2 + a)*C*b^2*c^4/(b*c^2* d^5 + a*d^7) - 3/2*sqrt(b*x^2 + a)*C*b^2*c^3*x/(b*c^2*d^4 + a*d^6) + 1/2*( b*x^2 + a)^(5/2)*D*c^3/(b*c^2*d^4*x^2 + a*d^6*x^2 + 2*b*c^3*d^3*x + 2*a*c* d^5*x + b*c^4*d^2 + a*c^2*d^4) + 1/2*(b*x^2 + a)^(3/2)*C*b*c^3/(b*c^2*d^4* x + a*d^6*x + b*c^3*d^3 + a*c*d^5) - 1/2*(b*x^2 + a)^(3/2)*D*b*c^3/(b*c^2* d^4 + a*d^6) - 3/2*sqrt(b*x^2 + a)*B*b^2*c^3/(b*c^2*d^4 + a*d^6) + 3/2*sqr t(b*x^2 + a)*B*b^2*c^2*x/(b*c^2*d^3 + a*d^5) - 1/2*(b*x^2 + a)^(5/2)*C*c^2 /(b*c^2*d^3*x^2 + a*d^5*x^2 + 2*b*c^3*d^2*x + 2*a*c*d^4*x + b*c^4*d + a*c^ 2*d^3) - 1/2*(b*x^2 + a)^(3/2)*B*b*c^2/(b*c^2*d^3*x + a*d^5*x + b*c^3*d^2 + a*c*d^4) + 1/2*(b*x^2 + a)^(3/2)*C*b*c^2/(b*c^2*d^3 + a*d^5) + 3/2*sqrt( b*x^2 + a)*A*b^2*c^2/(b*c^2*d^3 + a*d^5) - 3/2*sqrt(b*x^2 + a)*A*b^2*c*x/( b*c^2*d^2 + a*d^4) + 1/2*(b*x^2 + a)^(5/2)*B*c/(b*c^2*d^2*x^2 + a*d^4*x^2 + 2*b*c^3*d*x + 2*a*c*d^3*x + b*c^4 + a*c^2*d^2) + 1/2*(b*x^2 + a)^(3/2)*A *b*c/(b*c^2*d^2*x + a*d^4*x + b*c^3*d + a*c*d^3) - 1/2*(b*x^2 + a)^(3/2)*B *b*c/(b*c^2*d^2 + a*d^4) - 3*(b*x^2 + a)^(3/2)*D*c^2/(d^5*x + c*d^4) - 1/2 *(b*x^2 + a)^(5/2)*A/(b*c^2*d*x^2 + a*d^3*x^2 + 2*b*c^3*x + 2*a*c*d^2*x + b*c^4/d + a*c^2*d) + 1/2*(b*x^2 + a)^(3/2)*A*b/(b*c^2*d + a*d^3) + 2*(b*x^ 2 + a)^(3/2)*C*c/(d^4*x + c*d^3) - (b*x^2 + a)^(3/2)*B/(d^3*x + c*d^2) ...
Leaf count of result is larger than twice the leaf count of optimal. 1414 vs. \(2 (510) = 1020\).
Time = 0.68 (sec) , antiderivative size = 1414, normalized size of antiderivative = 2.58 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x, algorithm="giac ")
Output:
1/24*sqrt(b*x^2 + a)*((2*x*(3*D*b*x/d^3 - 4*(3*D*b^3*c*d^21 - C*b^3*d^22)/ (b^2*d^25)) + 3*(24*D*b^3*c^2*d^20 - 12*C*b^3*c*d^21 + 5*D*a*b^2*d^22 + 4* B*b^3*d^22)/(b^2*d^25))*x - 8*(30*D*b^3*c^3*d^19 - 18*C*b^3*c^2*d^20 + 12* D*a*b^2*c*d^21 + 9*B*b^3*c*d^21 - 4*C*a*b^2*d^22 - 3*A*b^3*d^22)/(b^2*d^25 )) - (30*D*b^2*c^5 - 20*C*b^2*c^4*d + 33*D*a*b*c^3*d^2 + 12*B*b^2*c^3*d^2 - 19*C*a*b*c^2*d^3 - 6*A*b^2*c^2*d^3 + 6*D*a^2*c*d^4 + 9*B*a*b*c*d^4 - 2*C *a^2*d^5 - 3*A*a*b*d^5)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b) *c)/sqrt(-b*c^2 - a*d^2))/(sqrt(-b*c^2 - a*d^2)*d^7) - 1/8*(120*D*b^2*c^4 - 80*C*b^2*c^3*d + 72*D*a*b*c^2*d^2 + 48*B*b^2*c^2*d^2 - 36*C*a*b*c*d^3 - 24*A*b^2*c*d^3 + 3*D*a^2*d^4 + 12*B*a*b*d^4)*log(abs(-sqrt(b)*x + sqrt(b*x ^2 + a)))/(sqrt(b)*d^7) - (12*(sqrt(b)*x - sqrt(b*x^2 + a))^3*D*b^2*c^5*d - 10*(sqrt(b)*x - sqrt(b*x^2 + a))^3*C*b^2*c^4*d^2 + 7*(sqrt(b)*x - sqrt(b *x^2 + a))^3*D*a*b*c^3*d^3 + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*b^2*c^3*d ^3 - 5*(sqrt(b)*x - sqrt(b*x^2 + a))^3*C*a*b*c^2*d^4 - 6*(sqrt(b)*x - sqrt (b*x^2 + a))^3*A*b^2*c^2*d^4 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a*b*c*d ^5 - (sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a*b*d^6 + 22*(sqrt(b)*x - sqrt(b*x^ 2 + a))^2*D*b^(5/2)*c^6 - 18*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*b^(5/2)*c^5 *d + (sqrt(b)*x - sqrt(b*x^2 + a))^2*D*a*b^(3/2)*c^4*d^2 + 14*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*b^(5/2)*c^4*d^2 + (sqrt(b)*x - sqrt(b*x^2 + a))^2*C* a*b^(3/2)*c^3*d^3 - 10*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*b^(5/2)*c^3*d^...
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^3,x)
Output:
int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^3, x)
Time = 0.18 (sec) , antiderivative size = 4155, normalized size of antiderivative = 7.58 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^3,x)
Output:
( - 72*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c**2*d**4 - 144*sqrt(a*d**2 + b*c**2)*log( - sqr t(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c*d**5*x - 72 *sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*d**6*x**2 + 96*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b *x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c**3*d**4 + 192*sqrt(a* d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x )*a**2*b*c**2*d**5*x + 96*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sq rt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c*d**6*x**2 - 144*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b** 3*c**4*d**2 - 288*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d** 2 + b*c**2) - a*d + b*c*x)*a*b**3*c**3*d**3*x + 216*sqrt(a*d**2 + b*c**2)* log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c**3*d **3 - 144*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c* *2) - a*d + b*c*x)*a*b**3*c**2*d**4*x**2 + 432*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c**2*d**4*x + 216*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c*d**5*x**2 + 336*sqrt(a*d**2 + b*c**2)*log( - sqrt (a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**5*d**2 + 672*s qrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*...