Integrand size = 37, antiderivative size = 456 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\left (a \left (B+\frac {a D}{b}\right )+(A b+a C) x\right ) (c+d x)^{3/2}}{a b \sqrt {a-b x^2}}+\frac {(15 A b d+25 a C d+6 a c D) \sqrt {c+d x} \sqrt {a-b x^2}}{15 a b^2}+\frac {2 D (c+d x)^{3/2} \sqrt {a-b x^2}}{5 b^2}+\frac {\left (15 A b^2 c d+a \left (63 a d^2 D+b \left (55 c C d+45 B d^2+6 c^2 D\right )\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {a} b^{5/2} d \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) (15 A b d+25 a C d+6 a c D) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{15 \sqrt {a} b^{5/2} d \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
(a*(B+a*D/b)+(A*b+C*a)*x)*(d*x+c)^(3/2)/a/b/(-b*x^2+a)^(1/2)+1/15*(15*A*b* d+25*C*a*d+6*D*a*c)*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/a/b^2+2/5*D*(d*x+c)^(3/ 2)*(-b*x^2+a)^(1/2)/b^2+1/15*(15*A*b^2*c*d+a*(63*a*d^2*D+b*(45*B*d^2+55*C* c*d+6*D*c^2)))*(d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticE(1/2*(1-b^(1/2) *x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2)) /a^(1/2)/b^(5/2)/d/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)- 1/15*(-a*d^2+b*c^2)*(15*A*b*d+25*C*a*d+6*D*a*c)*((d*x+c)/(c+a^(1/2)*d/b^(1 /2)))^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2) *2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/a^(1/2)/b^(5/2)/ d/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 28.12 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.43 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {\sqrt {a-b x^2} \left (\frac {(c+d x) \left (15 A b^2 c x+a^2 (25 C d+3 D (9 c+7 d x))+a b (15 A d+15 B (c+d x)+x (3 c (5 C-4 D x)-2 d x (5 C+3 D x)))\right )}{a b^2 \left (a-b x^2\right )}-\frac {d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (15 A b^2 c d+a \left (63 a d^2 D+b \left (55 c C d+45 B d^2+6 c^2 D\right )\right )\right ) \left (a-b x^2\right )+i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (15 A b^2 c d+a \left (63 a d^2 D+b \left (55 c C d+45 B d^2+6 c^2 D\right )\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+i \sqrt {a} \sqrt {b} d \left (\sqrt {b} c-\sqrt {a} d\right ) \left (15 A b^{3/2} d-15 \sqrt {a} b (2 c C+3 B d)-63 a^{3/2} d D+a \sqrt {b} (25 C d+6 c D)\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{a b^3 d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{15 \sqrt {c+d x}} \] Input:
Integrate[((c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(a - b*x^2)^(3/2),x]
Output:
(Sqrt[a - b*x^2]*(((c + d*x)*(15*A*b^2*c*x + a^2*(25*C*d + 3*D*(9*c + 7*d* x)) + a*b*(15*A*d + 15*B*(c + d*x) + x*(3*c*(5*C - 4*D*x) - 2*d*x*(5*C + 3 *D*x)))))/(a*b^2*(a - b*x^2)) - (d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(15*A* b^2*c*d + a*(63*a*d^2*D + b*(55*c*C*d + 45*B*d^2 + 6*c^2*D)))*(a - b*x^2) + I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(15*A*b^2*c*d + a*(63*a*d^2*D + b*(55* c*C*d + 45*B*d^2 + 6*c^2*D)))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sq rt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*A rcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a] *d)/(Sqrt[b]*c - Sqrt[a]*d)] + I*Sqrt[a]*Sqrt[b]*d*(Sqrt[b]*c - Sqrt[a]*d) *(15*A*b^(3/2)*d - 15*Sqrt[a]*b*(2*c*C + 3*B*d) - 63*a^(3/2)*d*D + a*Sqrt[ b]*(25*C*d + 6*c*D))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sq rt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sq rt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt [b]*c - Sqrt[a]*d)])/(a*b^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2 ))))/(15*Sqrt[c + d*x])
Time = 1.03 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.351, Rules used = {2176, 27, 2185, 27, 687, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2176 |
| \(\displaystyle \frac {\int -\frac {\sqrt {c+d x} \left (2 a d D x^2+(3 A b d+5 a C d+2 a c D) x+\frac {a (2 b c C+3 b B d+3 a d D)}{b}\right )}{2 \sqrt {a-b x^2}}dx}{a b}+\frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\int \frac {\sqrt {c+d x} \left (2 a d D x^2+(3 A b d+5 a C d+2 a c D) x+\frac {a (2 b c C+3 b B d+3 a d D)}{b}\right )}{\sqrt {a-b x^2}}dx}{2 a b}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {-\frac {2 \int -\frac {d^2 \sqrt {c+d x} (a (10 b c C+15 b B d+21 a d D)+b (15 A b d+25 a C d+6 a c D) x)}{2 \sqrt {a-b x^2}}dx}{5 b d^2}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\int \frac {\sqrt {c+d x} (a (10 b c C+15 b B d+21 a d D)+b (15 A b d+25 a C d+6 a c D) x)}{\sqrt {a-b x^2}}dx}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {-\frac {2 \int -\frac {b \left (a \left (15 b \left (2 C c^2+3 B d c+A d^2\right )+a d (25 C d+69 c D)\right )+\left (15 A c d b^2+a \left (63 a D d^2+b \left (6 D c^2+55 C d c+45 B d^2\right )\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b}-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{3} \int \frac {a \left (15 b \left (2 C c^2+3 B d c+A d^2\right )+a d (25 C d+69 c D)\right )+\left (15 A c d b^2+a \left (63 a D d^2+b \left (6 D c^2+55 C d c+45 B d^2\right )\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{3} \left (\frac {\left (a \left (63 a d^2 D+b \left (45 B d^2+6 c^2 D+55 c C d\right )\right )+15 A b^2 c d\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}-\frac {\left (b c^2-a d^2\right ) (6 a c D+25 a C d+15 A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{3} \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (a \left (63 a d^2 D+b \left (45 B d^2+6 c^2 D+55 c C d\right )\right )+15 A b^2 c d\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\left (b c^2-a d^2\right ) (6 a c D+25 a C d+15 A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{3} \left (-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a \left (63 a d^2 D+b \left (45 B d^2+6 c^2 D+55 c C d\right )\right )+15 A b^2 c d\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {\left (b c^2-a d^2\right ) (6 a c D+25 a C d+15 A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{3} \left (-\frac {\left (b c^2-a d^2\right ) (6 a c D+25 a C d+15 A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (63 a d^2 D+b \left (45 B d^2+6 c^2 D+55 c C d\right )\right )+15 A b^2 c d\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{3} \left (-\frac {\sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) (6 a c D+25 a C d+15 A b d) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (63 a d^2 D+b \left (45 B d^2+6 c^2 D+55 c C d\right )\right )+15 A b^2 c d\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} (6 a c D+25 a C d+15 A b d) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (63 a d^2 D+b \left (45 B d^2+6 c^2 D+55 c C d\right )\right )+15 A b^2 c d\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {(c+d x)^{3/2} \left (x (a C+A b)+a \left (\frac {a D}{b}+B\right )\right )}{a b \sqrt {a-b x^2}}-\frac {\frac {\frac {1}{3} \left (\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \left (b c^2-a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} (6 a c D+25 a C d+15 A b d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) \left (a \left (63 a d^2 D+b \left (45 B d^2+6 c^2 D+55 c C d\right )\right )+15 A b^2 c d\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {2}{3} \sqrt {a-b x^2} \sqrt {c+d x} (6 a c D+25 a C d+15 A b d)}{5 b}-\frac {4 a D \sqrt {a-b x^2} (c+d x)^{3/2}}{5 b}}{2 a b}\) |
Input:
Int[((c + d*x)^(3/2)*(A + B*x + C*x^2 + D*x^3))/(a - b*x^2)^(3/2),x]
Output:
((a*(B + (a*D)/b) + (A*b + a*C)*x)*(c + d*x)^(3/2))/(a*b*Sqrt[a - b*x^2]) - ((-4*a*D*(c + d*x)^(3/2)*Sqrt[a - b*x^2])/(5*b) + ((-2*(15*A*b*d + 25*a* C*d + 6*a*c*D)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 + ((-2*Sqrt[a]*(15*A*b^2*c *d + a*(63*a*d^2*D + b*(55*c*C*d + 45*B*d^2 + 6*c^2*D)))*Sqrt[c + d*x]*Sqr t[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqr t[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (2*Sqrt[a]*(b*c^2 - a*d^2)*(15*A*b *d + 25*a*C*d + 6*a*c*D)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)] *Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2 ]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b* x^2]))/3)/(5*b))/(2*a*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a + b*x^2, x], R = Coeff[PolynomialRema inder[Pq, a + b*x^2, x], x, 0], S = Coeff[PolynomialRemainder[Pq, a + b*x^2 , x], x, 1]}, Simp[(d + e*x)^m*(a + b*x^2)^(p + 1)*((a*S - b*R*x)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(d + e*x)*Qx - a*e*S*m + b*d*R*(2*p + 3) + b*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && !(IGtQ[m, 0] && R ationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Leaf count of result is larger than twice the leaf count of optimal. \(998\) vs. \(2(386)=772\).
Time = 5.21 (sec) , antiderivative size = 999, normalized size of antiderivative = 2.19
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (-b \,x^{2}+a \right )}\, \left (-\frac {2 \left (-b d x -b c \right ) \left (\frac {\left (A \,b^{2} c +B a b d +C a b c +a^{2} d D\right ) x}{2 a \,b^{3}}+\frac {A b d +B b c +C a d +D a c}{2 b^{3}}\right )}{\sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d x -b c \right )}}+\frac {2 D d x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b^{2}}-\frac {2 \left (-\frac {d \left (C d +2 D c \right )}{b}+\frac {4 D d c}{5 b}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (-\frac {A b \,d^{2}+2 B b c d +C a \,d^{2}+C b \,c^{2}+2 a c d D}{b^{2}}+\frac {A a b \,d^{2}+A \,b^{2} c^{2}+2 a b B c d +a^{2} C \,d^{2}+C a b \,c^{2}+2 a^{2} c d D}{b^{2} a}-\frac {d \left (A b d +B b c +C a d +D a c \right )}{2 b^{2}}-\frac {c \left (A \,b^{2} c +B a b d +C a b c +a^{2} d D\right )}{b^{2} a}-\frac {2 D d a c}{5 b^{2}}+\frac {\left (-\frac {d \left (C d +2 D c \right )}{b}+\frac {4 D d c}{5 b}\right ) a}{3 b}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (-\frac {B b \,d^{2}+2 C b c d +a \,d^{2} D+D b \,c^{2}}{b^{2}}-\frac {\left (A \,b^{2} c +B a b d +C a b c +a^{2} d D\right ) d}{2 a \,b^{2}}-\frac {3 D d^{2} a}{5 b^{2}}-\frac {2 \left (-\frac {d \left (C d +2 D c \right )}{b}+\frac {4 D d c}{5 b}\right ) c}{3 d}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}\) | \(999\) |
| default | \(\text {Expression too large to display}\) | \(3310\) |
Input:
int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x,method=_RETURNVER BOSE)
Output:
1/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)*((d*x+c)*(-b*x^2+a))^(1/2)*(-2*(-b*d*x-b* c)*(1/2*(A*b^2*c+B*a*b*d+C*a*b*c+D*a^2*d)/a/b^3*x+1/2*(A*b*d+B*b*c+C*a*d+D *a*c)/b^3)/((x^2-a/b)*(-b*d*x-b*c))^(1/2)+2/5*D*d/b^2*x*(-b*d*x^3-b*c*x^2+ a*d*x+a*c)^(1/2)-2/3*(-1/b*d*(C*d+2*D*c)+4/5*D*d/b*c)/b/d*(-b*d*x^3-b*c*x^ 2+a*d*x+a*c)^(1/2)+2*(-(A*b*d^2+2*B*b*c*d+C*a*d^2+C*b*c^2+2*D*a*c*d)/b^2+1 /b^2*(A*a*b*d^2+A*b^2*c^2+2*B*a*b*c*d+C*a^2*d^2+C*a*b*c^2+2*D*a^2*c*d)/a-1 /2/b^2*d*(A*b*d+B*b*c+C*a*d+D*a*c)-1/b^2*c*(A*b^2*c+B*a*b*d+C*a*b*c+D*a^2* d)/a-2/5*D*d/b^2*a*c+1/3*(-1/b*d*(C*d+2*D*c)+4/5*D*d/b*c)/b*a)*(c/d-1/b*(a *b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/ d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^(1/ 2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1 /2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(-(B* b*d^2+2*C*b*c*d+D*a*d^2+D*b*c^2)/b^2-1/2*(A*b^2*c+B*a*b*d+C*a*b*c+D*a^2*d) *d/a/b^2-3/5*D*d^2/b^2*a-2/3*(-1/b*d*(C*d+2*D*c)+4/5*D*d/b*c)/d*c)*(c/d-1/ b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/ (-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2))) ^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((-c/d-1/b*(a*b)^(1/2))*Elliptic E(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b* (a*b)^(1/2)))^(1/2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/ 2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))))
Time = 0.09 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.29 \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {{\left (6 \, D a^{2} b c^{3} - 5 \, {\left (7 \, C a^{2} b - 3 \, A a b^{2}\right )} c^{2} d - 18 \, {\left (8 \, D a^{3} + 5 \, B a^{2} b\right )} c d^{2} - 15 \, {\left (5 \, C a^{3} + 3 \, A a^{2} b\right )} d^{3} - {\left (6 \, D a b^{2} c^{3} - 5 \, {\left (7 \, C a b^{2} - 3 \, A b^{3}\right )} c^{2} d - 18 \, {\left (8 \, D a^{2} b + 5 \, B a b^{2}\right )} c d^{2} - 15 \, {\left (5 \, C a^{2} b + 3 \, A a b^{2}\right )} d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (6 \, D a^{2} b c^{2} d + 5 \, {\left (11 \, C a^{2} b + 3 \, A a b^{2}\right )} c d^{2} + 9 \, {\left (7 \, D a^{3} + 5 \, B a^{2} b\right )} d^{3} - {\left (6 \, D a b^{2} c^{2} d + 5 \, {\left (11 \, C a b^{2} + 3 \, A b^{3}\right )} c d^{2} + 9 \, {\left (7 \, D a^{2} b + 5 \, B a b^{2}\right )} d^{3}\right )} x^{2}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (6 \, D a b^{2} d^{3} x^{3} - 3 \, {\left (9 \, D a^{2} b + 5 \, B a b^{2}\right )} c d^{2} - 5 \, {\left (5 \, C a^{2} b + 3 \, A a b^{2}\right )} d^{3} + 2 \, {\left (6 \, D a b^{2} c d^{2} + 5 \, C a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (5 \, {\left (C a b^{2} + A b^{3}\right )} c d^{2} + {\left (7 \, D a^{2} b + 5 \, B a b^{2}\right )} d^{3}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}}{45 \, {\left (a b^{4} d^{2} x^{2} - a^{2} b^{3} d^{2}\right )}} \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x, algorithm= "fricas")
Output:
1/45*((6*D*a^2*b*c^3 - 5*(7*C*a^2*b - 3*A*a*b^2)*c^2*d - 18*(8*D*a^3 + 5*B *a^2*b)*c*d^2 - 15*(5*C*a^3 + 3*A*a^2*b)*d^3 - (6*D*a*b^2*c^3 - 5*(7*C*a*b ^2 - 3*A*b^3)*c^2*d - 18*(8*D*a^2*b + 5*B*a*b^2)*c*d^2 - 15*(5*C*a^2*b + 3 *A*a*b^2)*d^3)*x^2)*sqrt(-b*d)*weierstrassPInverse(4/3*(b*c^2 + 3*a*d^2)/( b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d) + 3*(6*D*a^2 *b*c^2*d + 5*(11*C*a^2*b + 3*A*a*b^2)*c*d^2 + 9*(7*D*a^3 + 5*B*a^2*b)*d^3 - (6*D*a*b^2*c^2*d + 5*(11*C*a*b^2 + 3*A*b^3)*c*d^2 + 9*(7*D*a^2*b + 5*B*a *b^2)*d^3)*x^2)*sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^2 + 3*a*d^ 2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d)) + 3*(6* D*a*b^2*d^3*x^3 - 3*(9*D*a^2*b + 5*B*a*b^2)*c*d^2 - 5*(5*C*a^2*b + 3*A*a*b ^2)*d^3 + 2*(6*D*a*b^2*c*d^2 + 5*C*a*b^2*d^3)*x^2 - 3*(5*(C*a*b^2 + A*b^3) *c*d^2 + (7*D*a^2*b + 5*B*a*b^2)*d^3)*x)*sqrt(-b*x^2 + a)*sqrt(d*x + c))/( a*b^4*d^2*x^2 - a^2*b^3*d^2)
Timed out. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**(3/2)*(D*x**3+C*x**2+B*x+A)/(-b*x**2+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x, algorithm= "maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^(3/2)/(-b*x^2 + a)^(3/2), x)
\[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (D x^{3} + C x^{2} + B x + A\right )} {\left (d x + c\right )}^{\frac {3}{2}}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x, algorithm= "giac")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)*(d*x + c)^(3/2)/(-b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(3/2),x)
Output:
int(((c + d*x)^(3/2)*(A + B*x + C*x^2 + x^3*D))/(a - b*x^2)^(3/2), x)
\[ \int \frac {(c+d x)^{3/2} \left (A+B x+C x^2+D x^3\right )}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {\left (d x +c \right )^{\frac {3}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x \] Input:
int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x)
Output:
int((d*x+c)^(3/2)*(D*x^3+C*x^2+B*x+A)/(-b*x^2+a)^(3/2),x)