\(\int \frac {\sqrt {a+b x^2} (A+B x+C x^2+D x^3)}{(c+d x)^4} \, dx\) [66]

Optimal result

Integrand size = 34, antiderivative size = 449 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\frac {D \sqrt {a+b x^2}}{d^4}+\frac {\left (a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b \left (c^3 C d-A c d^3-2 c^4 D\right )\right ) \sqrt {a+b x^2}}{2 d^4 \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {\left (2 a^2 d^4 (C d-3 c D)+a b c d^2 \left (6 c C d-B d^2-15 c^2 D\right )+b^2 \left (3 c^4 C d-A c^2 d^3-8 c^5 D\right )\right ) \sqrt {a+b x^2}}{2 d^4 \left (b c^2+a d^2\right )^2 (c+d x)}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (a+b x^2\right )^{3/2}}{3 d^2 \left (b c^2+a d^2\right ) (c+d x)^3}+\frac {\sqrt {b} (C d-4 c D) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^5}-\frac {\left (2 a^3 d^6 D-2 b^3 c^5 (C d-4 c D)-a^2 b d^4 \left (4 c C d-B d^2-15 c^2 D\right )-a b^2 c d^2 \left (5 c^2 C d-A d^3-20 c^3 D\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^5 \left (b c^2+a d^2\right )^{5/2}} \] Output:

D*(b*x^2+a)^(1/2)/d^4+1/2*(a*d^2*(-B*d^2+2*C*c*d-3*D*c^2)+b*(-A*c*d^3+C*c^ 
3*d-2*D*c^4))*(b*x^2+a)^(1/2)/d^4/(a*d^2+b*c^2)/(d*x+c)^2-1/2*(2*a^2*d^4*( 
C*d-3*D*c)+a*b*c*d^2*(-B*d^2+6*C*c*d-15*D*c^2)+b^2*(-A*c^2*d^3+3*C*c^4*d-8 
*D*c^5))*(b*x^2+a)^(1/2)/d^4/(a*d^2+b*c^2)^2/(d*x+c)-1/3*(A*d^3-B*c*d^2+C* 
c^2*d-D*c^3)*(b*x^2+a)^(3/2)/d^2/(a*d^2+b*c^2)/(d*x+c)^3+b^(1/2)*(C*d-4*D* 
c)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/d^5-1/2*(2*a^3*d^6*D-2*b^3*c^5*(C*d- 
4*D*c)-a^2*b*d^4*(-B*d^2+4*C*c*d-15*D*c^2)-a*b^2*c*d^2*(-A*d^3+5*C*c^2*d-2 
0*D*c^3))*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^5/(a 
*d^2+b*c^2)^(5/2)
 

Mathematica [A] (verified)

Time = 11.54 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\frac {-\frac {d \sqrt {a+b x^2} \left (2 \left (b c^2+a d^2\right )^2 \left (c^2 C d-B c d^2+A d^3-c^3 D\right )+\left (b c^2+a d^2\right ) \left (3 a d^2 \left (-2 c C d+B d^2+3 c^2 D\right )+b c \left (-7 c^2 C d+4 B c d^2-A d^3+10 c^3 D\right )\right ) (c+d x)-\left (-6 a^2 d^4 (C d-3 c D)+b^2 c^2 \left (-11 c^2 C d+2 B c d^2+A d^3+26 c^3 D\right )+a b d^2 \left (-20 c^2 C d+5 B c d^2-2 A d^3+47 c^3 D\right )\right ) (c+d x)^2-6 \left (b c^2+a d^2\right )^2 D (c+d x)^3\right )}{\left (b c^2+a d^2\right )^2 (c+d x)^3}+\frac {3 \left (2 a^3 d^6 D+2 b^3 c^5 (-C d+4 c D)+a^2 b d^4 \left (-4 c C d+B d^2+15 c^2 D\right )+a b^2 c d^2 \left (-5 c^2 C d+A d^3+20 c^3 D\right )\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{5/2}}+6 \sqrt {b} (C d-4 c D) \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )-\frac {3 \left (2 a^3 d^6 D+2 b^3 c^5 (-C d+4 c D)+a^2 b d^4 \left (-4 c C d+B d^2+15 c^2 D\right )+a b^2 c d^2 \left (-5 c^2 C d+A d^3+20 c^3 D\right )\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{5/2}}}{6 d^5} \] Input:

Integrate[(Sqrt[a + b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^4,x]
 

Output:

(-((d*Sqrt[a + b*x^2]*(2*(b*c^2 + a*d^2)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^ 
3*D) + (b*c^2 + a*d^2)*(3*a*d^2*(-2*c*C*d + B*d^2 + 3*c^2*D) + b*c*(-7*c^2 
*C*d + 4*B*c*d^2 - A*d^3 + 10*c^3*D))*(c + d*x) - (-6*a^2*d^4*(C*d - 3*c*D 
) + b^2*c^2*(-11*c^2*C*d + 2*B*c*d^2 + A*d^3 + 26*c^3*D) + a*b*d^2*(-20*c^ 
2*C*d + 5*B*c*d^2 - 2*A*d^3 + 47*c^3*D))*(c + d*x)^2 - 6*(b*c^2 + a*d^2)^2 
*D*(c + d*x)^3))/((b*c^2 + a*d^2)^2*(c + d*x)^3)) + (3*(2*a^3*d^6*D + 2*b^ 
3*c^5*(-(C*d) + 4*c*D) + a^2*b*d^4*(-4*c*C*d + B*d^2 + 15*c^2*D) + a*b^2*c 
*d^2*(-5*c^2*C*d + A*d^3 + 20*c^3*D))*Log[c + d*x])/(b*c^2 + a*d^2)^(5/2) 
+ 6*Sqrt[b]*(C*d - 4*c*D)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]] - (3*(2*a^3*d 
^6*D + 2*b^3*c^5*(-(C*d) + 4*c*D) + a^2*b*d^4*(-4*c*C*d + B*d^2 + 15*c^2*D 
) + a*b^2*c*d^2*(-5*c^2*C*d + A*d^3 + 20*c^3*D))*Log[a*d - b*c*x + Sqrt[b* 
c^2 + a*d^2]*Sqrt[a + b*x^2]])/(b*c^2 + a*d^2)^(5/2))/(6*d^5)
 

Rubi [A] (verified)

Time = 1.17 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2182, 27, 2182, 25, 27, 681, 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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx\)

\(\Big \downarrow \) 2182

\(\displaystyle -\frac {\int -\frac {3 \sqrt {b x^2+a} \left (\left (\frac {b c^2}{d}+a d\right ) D x^2+\frac {\left (b c^2+a d^2\right ) (C d-c D) x}{d^2}+A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )}{(c+d x)^3}dx}{3 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left (\left (\frac {b c^2}{d}+a d\right ) D x^2+\frac {\left (b c^2+a d^2\right ) (C d-c D) x}{d^2}+A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )}{(c+d x)^3}dx}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {-\frac {\int -\frac {\left (2 \left (A b^2 c^2+a \left (-\frac {b D c^3}{d}+b B d c+a d (C d-2 c D)\right )\right ) d^2+\left (2 a^2 D d^4-a b \left (-7 D c^2+2 C d c-B d^2\right ) d^2-b^2 \left (-4 D c^4+C d c^3-A d^3 c\right )\right ) x\right ) \sqrt {b x^2+a}}{d^2 (c+d x)^2}dx}{2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (2 \left (A b^2 c^2+a \left (-\frac {b D c^3}{d}+b B d c+a d (C d-2 c D)\right )\right ) d^2+\left (2 a^2 D d^4-a b \left (-7 D c^2+2 C d c-B d^2\right ) d^2-b^2 \left (-4 D c^4+C d c^3-A d^3 c\right )\right ) x\right ) \sqrt {b x^2+a}}{d^2 (c+d x)^2}dx}{2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (2 \left (A b^2 c^2+a \left (-\frac {b D c^3}{d}+b B d c+a d (C d-2 c D)\right )\right ) d^2+\left (2 a^2 D d^4-a b \left (-7 D c^2+2 C d c-B d^2\right ) d^2-b^2 \left (-4 D c^4+C d c^3-A d^3 c\right )\right ) x\right ) \sqrt {b x^2+a}}{(c+d x)^2}dx}{2 d^2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \left (2 b (C d-4 c D) x \left (b c^2+a d^2\right )^2+a d \left (2 a^2 D d^4-a b \left (-7 D c^2+2 C d c-B d^2\right ) d^2-b^2 \left (-4 D c^4+C d c^3-A d^3 c\right )\right )\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}-\frac {\sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )^2 (C d-4 c D)-d x \left (2 a^2 d^4 D-a b d^2 \left (-B d^2-7 c^2 D+2 c C d\right )-b^2 \left (-A c d^3-4 c^4 D+c^3 C d\right )\right )\right )}{d^2 (c+d x)}}{2 d^2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {2 b (C d-4 c D) x \left (b c^2+a d^2\right )^2+a d \left (2 a^2 D d^4-a b \left (-7 D c^2+2 C d c-B d^2\right ) d^2-b^2 \left (-4 D c^4+C d c^3-A d^3 c\right )\right )}{(c+d x) \sqrt {b x^2+a}}dx}{d^2}-\frac {\sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )^2 (C d-4 c D)-d x \left (2 a^2 d^4 D-a b d^2 \left (-B d^2-7 c^2 D+2 c C d\right )-b^2 \left (-A c d^3-4 c^4 D+c^3 C d\right )\right )\right )}{d^2 (c+d x)}}{2 d^2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\frac {\frac {\frac {\left (2 a^3 d^6 D-a^2 b d^4 \left (-B d^2-15 c^2 D+4 c C d\right )-a b^2 c d^2 \left (-A d^3-20 c^3 D+5 c^2 C d\right )-2 b^3 c^5 (C d-4 c D)\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {2 b \left (a d^2+b c^2\right )^2 (C d-4 c D) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )^2 (C d-4 c D)-d x \left (2 a^2 d^4 D-a b d^2 \left (-B d^2-7 c^2 D+2 c C d\right )-b^2 \left (-A c d^3-4 c^4 D+c^3 C d\right )\right )\right )}{d^2 (c+d x)}}{2 d^2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {\left (2 a^3 d^6 D-a^2 b d^4 \left (-B d^2-15 c^2 D+4 c C d\right )-a b^2 c d^2 \left (-A d^3-20 c^3 D+5 c^2 C d\right )-2 b^3 c^5 (C d-4 c D)\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {2 b \left (a d^2+b c^2\right )^2 (C d-4 c D) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )^2 (C d-4 c D)-d x \left (2 a^2 d^4 D-a b d^2 \left (-B d^2-7 c^2 D+2 c C d\right )-b^2 \left (-A c d^3-4 c^4 D+c^3 C d\right )\right )\right )}{d^2 (c+d x)}}{2 d^2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {\left (2 a^3 d^6 D-a^2 b d^4 \left (-B d^2-15 c^2 D+4 c C d\right )-a b^2 c d^2 \left (-A d^3-20 c^3 D+5 c^2 C d\right )-2 b^3 c^5 (C d-4 c D)\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}+\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2 (C d-4 c D)}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )^2 (C d-4 c D)-d x \left (2 a^2 d^4 D-a b d^2 \left (-B d^2-7 c^2 D+2 c C d\right )-b^2 \left (-A c d^3-4 c^4 D+c^3 C d\right )\right )\right )}{d^2 (c+d x)}}{2 d^2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2 (C d-4 c D)}{d}-\frac {\left (2 a^3 d^6 D-a^2 b d^4 \left (-B d^2-15 c^2 D+4 c C d\right )-a b^2 c d^2 \left (-A d^3-20 c^3 D+5 c^2 C d\right )-2 b^3 c^5 (C d-4 c D)\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}}{d^2}-\frac {\sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )^2 (C d-4 c D)-d x \left (2 a^2 d^4 D-a b d^2 \left (-B d^2-7 c^2 D+2 c C d\right )-b^2 \left (-A c d^3-4 c^4 D+c^3 C d\right )\right )\right )}{d^2 (c+d x)}}{2 d^2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right )^2 (C d-4 c D)}{d}-\frac {\text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right ) \left (2 a^3 d^6 D-a^2 b d^4 \left (-B d^2-15 c^2 D+4 c C d\right )-a b^2 c d^2 \left (-A d^3-20 c^3 D+5 c^2 C d\right )-2 b^3 c^5 (C d-4 c D)\right )}{d \sqrt {a d^2+b c^2}}}{d^2}-\frac {\sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )^2 (C d-4 c D)-d x \left (2 a^2 d^4 D-a b d^2 \left (-B d^2-7 c^2 D+2 c C d\right )-b^2 \left (-A c d^3-4 c^4 D+c^3 C d\right )\right )\right )}{d^2 (c+d x)}}{2 d^2 \left (a d^2+b c^2\right )}-\frac {d \left (a+b x^2\right )^{3/2} \left (-a \left (-B d-\frac {3 c^2 D}{d}+2 c C\right )+A b c-\frac {b c^3 (C d-2 c D)}{d^3}\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\)

Input:

Int[(Sqrt[a + b*x^2]*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^4,x]
 

Output:

-1/3*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a + b*x^2)^(3/2))/(d^2*(b*c^2 + 
 a*d^2)*(c + d*x)^3) + (-1/2*(d*(A*b*c - (b*c^3*(C*d - 2*c*D))/d^3 - a*(2* 
c*C - B*d - (3*c^2*D)/d))*(a + b*x^2)^(3/2))/((b*c^2 + a*d^2)*(c + d*x)^2) 
 + (-(((2*(b*c^2 + a*d^2)^2*(C*d - 4*c*D) - d*(2*a^2*d^4*D - a*b*d^2*(2*c* 
C*d - B*d^2 - 7*c^2*D) - b^2*(c^3*C*d - A*c*d^3 - 4*c^4*D))*x)*Sqrt[a + b* 
x^2])/(d^2*(c + d*x))) + ((2*Sqrt[b]*(b*c^2 + a*d^2)^2*(C*d - 4*c*D)*ArcTa 
nh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/d - ((2*a^3*d^6*D - 2*b^3*c^5*(C*d - 4*c* 
D) - a^2*b*d^4*(4*c*C*d - B*d^2 - 15*c^2*D) - a*b^2*c*d^2*(5*c^2*C*d - A*d 
^3 - 20*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]))/d^2)/(2*d^2*(b*c^2 + a*d^2)))/(b*c^2 + a*d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 224

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]
 

rule 488

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]
 

rule 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 719

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]
 

rule 2182

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2733\) vs. \(2(421)=842\).

Time = 1.54 (sec) , antiderivative size = 2734, normalized size of antiderivative = 6.09

method	result	size
default	\(\text {Expression too large to display}\)	\(2734\)

Input:

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

D/d^4*((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*l 
n((-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)))+(C*d-3*D*c)/d^5*(-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)^(3/2)-b*c*d/(a* 
d^2+b*c^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)))+2*b/(a*d^2+b*c^2)*d^2*(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))))+(B*d^2-2*C 
*c*d+3*D*c^2)/d^6*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2*(b*(x+c/d)^2-2*b*c/d*( 
x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+1/2*b*c*d/(a*d^2+b*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)^(3/2)-b*c*d/(a 
*d^2+b*c^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^...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{4}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**4,x)
 

Output:

Integral(sqrt(a + b*x**2)*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**4, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2553 vs. \(2 (419) = 838\).

Time = 0.14 (sec) , antiderivative size = 2553, normalized size of antiderivative = 5.69 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x, algorithm="maxi 
ma")
 

Output:

1/2*sqrt(b*x^2 + a)*D*b^2*c^5/(b^2*c^4*d^5*x + 2*a*b*c^2*d^7*x + a^2*d^9*x 
 + b^2*c^5*d^4 + 2*a*b*c^3*d^6 + a^2*c*d^8) + 1/2*(b*x^2 + a)^(3/2)*D*b*c^ 
4/(b^2*c^4*d^4*x^2 + 2*a*b*c^2*d^6*x^2 + a^2*d^8*x^2 + 2*b^2*c^5*d^3*x + 4 
*a*b*c^3*d^5*x + 2*a^2*c*d^7*x + b^2*c^6*d^2 + 2*a*b*c^4*d^4 + a^2*c^2*d^6 
) - 1/2*sqrt(b*x^2 + a)*C*b^2*c^4/(b^2*c^4*d^4*x + 2*a*b*c^2*d^6*x + a^2*d 
^8*x + b^2*c^5*d^3 + 2*a*b*c^3*d^5 + a^2*c*d^7) - 1/2*sqrt(b*x^2 + a)*D*b^ 
2*c^4/(b^2*c^4*d^4 + 2*a*b*c^2*d^6 + a^2*d^8) - 1/2*(b*x^2 + a)^(3/2)*C*b* 
c^3/(b^2*c^4*d^3*x^2 + 2*a*b*c^2*d^5*x^2 + a^2*d^7*x^2 + 2*b^2*c^5*d^2*x + 
 4*a*b*c^3*d^4*x + 2*a^2*c*d^6*x + b^2*c^6*d + 2*a*b*c^4*d^3 + a^2*c^2*d^5 
) + 1/2*sqrt(b*x^2 + a)*B*b^2*c^3/(b^2*c^4*d^3*x + 2*a*b*c^2*d^5*x + a^2*d 
^7*x + b^2*c^5*d^2 + 2*a*b*c^3*d^4 + a^2*c*d^6) + 1/2*sqrt(b*x^2 + a)*C*b^ 
2*c^3/(b^2*c^4*d^3 + 2*a*b*c^2*d^5 + a^2*d^7) + 1/2*(b*x^2 + a)^(3/2)*B*b* 
c^2/(b^2*c^4*d^2*x^2 + 2*a*b*c^2*d^4*x^2 + a^2*d^6*x^2 + 2*b^2*c^5*d*x + 4 
*a*b*c^3*d^3*x + 2*a^2*c*d^5*x + b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4) - 
1/2*sqrt(b*x^2 + a)*A*b^2*c^2/(b^2*c^4*d^2*x + 2*a*b*c^2*d^4*x + a^2*d^6*x 
 + b^2*c^5*d + 2*a*b*c^3*d^3 + a^2*c*d^5) - 1/2*sqrt(b*x^2 + a)*B*b^2*c^2/ 
(b^2*c^4*d^2 + 2*a*b*c^2*d^4 + a^2*d^6) + 1/3*(b*x^2 + a)^(3/2)*D*c^3/(b*c 
^2*d^5*x^3 + a*d^7*x^3 + 3*b*c^3*d^4*x^2 + 3*a*c*d^6*x^2 + 3*b*c^4*d^3*x + 
 3*a*c^2*d^5*x + b*c^5*d^2 + a*c^3*d^4) - 3/2*sqrt(b*x^2 + a)*D*b*c^3/(b*c 
^2*d^5*x + a*d^7*x + b*c^3*d^4 + a*c*d^6) - 1/2*(b*x^2 + a)^(3/2)*A*b*c...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2390 vs. \(2 (419) = 838\).

Time = 0.40 (sec) , antiderivative size = 2390, normalized size of antiderivative = 5.32 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x, algorithm="giac 
")
 

Output:

(8*D*b^3*c^6 - 2*C*b^3*c^5*d + 20*D*a*b^2*c^4*d^2 - 5*C*a*b^2*c^3*d^3 + 15 
*D*a^2*b*c^2*d^4 - 4*C*a^2*b*c*d^5 + A*a*b^2*c*d^5 + 2*D*a^3*d^6 + B*a^2*b 
*d^6)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - 
a*d^2))/((b^2*c^4*d^5 + 2*a*b*c^2*d^7 + a^2*d^9)*sqrt(-b*c^2 - a*d^2)) + s 
qrt(b*x^2 + a)*D/d^4 + 1/3*(36*(sqrt(b)*x - sqrt(b*x^2 + a))^5*D*b^3*c^6*d 
^2 - 18*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*b^3*c^5*d^3 + 66*(sqrt(b)*x - sq 
rt(b*x^2 + a))^5*D*a*b^2*c^4*d^4 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^5*B*b^3 
*c^4*d^4 - 33*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*a*b^2*c^3*d^5 + 27*(sqrt(b 
)*x - sqrt(b*x^2 + a))^5*D*a^2*b*c^2*d^6 + 12*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^5*B*a*b^2*c^2*d^6 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^5*C*a^2*b*c*d^7 - 3 
*(sqrt(b)*x - sqrt(b*x^2 + a))^5*A*a*b^2*c*d^7 + 3*(sqrt(b)*x - sqrt(b*x^2 
 + a))^5*B*a^2*b*d^8 + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D*b^(7/2)*c^7*d 
 - 54*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*b^(7/2)*c^6*d^2 + 192*(sqrt(b)*x - 
 sqrt(b*x^2 + a))^4*D*a*b^(5/2)*c^5*d^3 + 12*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^4*B*b^(7/2)*c^5*d^3 - 87*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a*b^(5/2)*c^4* 
d^4 + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*b^(7/2)*c^4*d^4 + 39*(sqrt(b)*x 
- sqrt(b*x^2 + a))^4*D*a^2*b^(3/2)*c^3*d^5 + 24*(sqrt(b)*x - sqrt(b*x^2 + 
a))^4*B*a*b^(5/2)*c^3*d^5 - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^2*b^(3/ 
2)*c^2*d^6 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a*b^(5/2)*c^2*d^6 - 18*(s 
qrt(b)*x - sqrt(b*x^2 + a))^4*D*a^3*sqrt(b)*c*d^7 - 3*(sqrt(b)*x - sqrt...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^4} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^4,x)
 

Output:

int(((a + b*x^2)^(1/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 3915, normalized size of antiderivative = 8.72 \[ \int \frac {\sqrt {a+b x^2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^4} \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^(1/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^4,x)
 

Output:

(6*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**3*c**3*d**6 + 18*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**3*c**2*d**7*x + 18*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**3*c*d**8 
*x**2 + 6*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**3*d**9*x**3 + 3*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**4*d**4 + 9*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**3*d**5*x + 3*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**3*d**5 + 9*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**6*x**2 + 9*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**6*x + 3*sqrt(a*d**2 + b*c**2)*log(sq 
rt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c*d**7*x**3 
+ 9*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*d**7*x**2 + 3*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**8*x**3 + 33*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**5*d**4 + 99*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**4*d**5*x + 99*sqrt(a*d**2 + b*c**2...