\(\int \frac {(a+b x^2)^{5/2} (A+B x+C x^2+D x^3)}{(c+d x)^{10}} \, dx\) [98]

Optimal result

Integrand size = 34, antiderivative size = 767 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=-\frac {5 a^2 b^2 \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (b^2 c^2 (c C-10 B d)-8 a^2 d^3 D-a b d \left (10 c C d-B d^2-3 c^2 D\right )\right )\right ) (a d-b c x) \sqrt {a+b x^2}}{128 \left (b c^2+a d^2\right )^6 (c+d x)^2}-\frac {5 a b \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (b^2 c^2 (c C-10 B d)-8 a^2 d^3 D-a b d \left (10 c C d-B d^2-3 c^2 D\right )\right )\right ) (a d-b c x) \left (a+b x^2\right )^{3/2}}{192 \left (b c^2+a d^2\right )^5 (c+d x)^4}-\frac {\left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (b^2 c^2 (c C-10 B d)-8 a^2 d^3 D-a b d \left (10 c C d-B d^2-3 c^2 D\right )\right )\right ) (a d-b c x) \left (a+b x^2\right )^{5/2}}{48 \left (b c^2+a d^2\right )^4 (c+d x)^6}-\frac {\left (c^2 C d-B c d^2+A d^3-c^3 D\right ) \left (a+b x^2\right )^{7/2}}{9 d^2 \left (b c^2+a d^2\right ) (c+d x)^9}+\frac {\left (9 a d^2 \left (2 c C d-B d^2-3 c^2 D\right )+b c \left (7 c^2 C d+2 B c d^2-11 A d^3-16 c^3 D\right )\right ) \left (a+b x^2\right )^{7/2}}{72 d^2 \left (b c^2+a d^2\right )^2 (c+d x)^8}-\frac {\left (72 a^2 d^4 (C d-3 c D)-b^2 c^2 \left (7 c^2 C d+2 B c d^2-83 A d^3+56 c^3 D\right )-a b d^2 \left (34 c^2 C d-97 B c d^2+16 A d^3+173 c^3 D\right )\right ) \left (a+b x^2\right )^{7/2}}{504 d^2 \left (b c^2+a d^2\right )^3 (c+d x)^7}-\frac {5 a^3 b^3 \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (b^2 c^2 (c C-10 B d)-8 a^2 d^3 D-a b d \left (10 c C d-B d^2-3 c^2 D\right )\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{128 \left (b c^2+a d^2\right )^{13/2}} \] Output:

-5/128*a^2*b^2*(A*b^2*c*(-3*a*d^2+8*b*c^2)-a*(b^2*c^2*(-10*B*d+C*c)-8*a^2* 
d^3*D-a*b*d*(-B*d^2+10*C*c*d-3*D*c^2)))*(-b*c*x+a*d)*(b*x^2+a)^(1/2)/(a*d^ 
2+b*c^2)^6/(d*x+c)^2-5/192*a*b*(A*b^2*c*(-3*a*d^2+8*b*c^2)-a*(b^2*c^2*(-10 
*B*d+C*c)-8*a^2*d^3*D-a*b*d*(-B*d^2+10*C*c*d-3*D*c^2)))*(-b*c*x+a*d)*(b*x^ 
2+a)^(3/2)/(a*d^2+b*c^2)^5/(d*x+c)^4-1/48*(A*b^2*c*(-3*a*d^2+8*b*c^2)-a*(b 
^2*c^2*(-10*B*d+C*c)-8*a^2*d^3*D-a*b*d*(-B*d^2+10*C*c*d-3*D*c^2)))*(-b*c*x 
+a*d)*(b*x^2+a)^(5/2)/(a*d^2+b*c^2)^4/(d*x+c)^6-1/9*(A*d^3-B*c*d^2+C*c^2*d 
-D*c^3)*(b*x^2+a)^(7/2)/d^2/(a*d^2+b*c^2)/(d*x+c)^9+1/72*(9*a*d^2*(-B*d^2+ 
2*C*c*d-3*D*c^2)+b*c*(-11*A*d^3+2*B*c*d^2+7*C*c^2*d-16*D*c^3))*(b*x^2+a)^( 
7/2)/d^2/(a*d^2+b*c^2)^2/(d*x+c)^8-1/504*(72*a^2*d^4*(C*d-3*D*c)-b^2*c^2*( 
-83*A*d^3+2*B*c*d^2+7*C*c^2*d+56*D*c^3)-a*b*d^2*(16*A*d^3-97*B*c*d^2+34*C* 
c^2*d+173*D*c^3))*(b*x^2+a)^(7/2)/d^2/(a*d^2+b*c^2)^3/(d*x+c)^7-5/128*a^3* 
b^3*(A*b^2*c*(-3*a*d^2+8*b*c^2)-a*(b^2*c^2*(-10*B*d+C*c)-8*a^2*d^3*D-a*b*d 
*(-B*d^2+10*C*c*d-3*D*c^2)))*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x 
^2+a)^(1/2))/(a*d^2+b*c^2)^(13/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1955\) vs. \(2(767)=1534\).

Time = 17.89 (sec) , antiderivative size = 1955, normalized size of antiderivative = 2.55 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx =\text {Too large to display} \] Input:

Integrate[((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^10,x]
 

Output:

Sqrt[a + b*x^2]*(-1/9*((b*c^2 + a*d^2)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3* 
D))/(d^8*(c + d*x)^9) - ((b*c^2 + a*d^2)*(-55*b*c^3*C*d + 46*b*B*c^2*d^2 - 
 37*A*b*c*d^3 - 18*a*c*C*d^3 + 9*a*B*d^4 + 64*b*c^4*D + 27*a*c^2*d^2*D))/( 
72*d^8*(c + d*x)^8) + (-1127*b^2*c^4*C*d + 758*b^2*B*c^3*d^2 - 461*A*b^2*c 
^2*d^3 - 890*a*b*c^2*C*d^3 + 449*a*b*B*c*d^4 - 152*a*A*b*d^5 - 72*a^2*C*d^ 
5 + 1568*b^2*c^5*D + 1475*a*b*c^3*d^2*D + 216*a^2*c*d^4*D)/(504*d^8*(c + d 
*x)^7) + (3626*b^3*c^5*C*d - 1844*b^3*B*c^4*d^2 + 758*A*b^3*c^3*d^3 + 5031 
*a*b^2*c^3*C*d^3 - 2196*a*b^2*B*c^2*d^4 + 753*a*A*b^2*c*d^5 + 1410*a^2*b*c 
*C*d^5 - 357*a^2*b*B*d^6 - 6272*b^3*c^6*D - 9762*a*b^2*c^4*d^2*D - 3663*a^ 
2*b*c^2*d^4*D - 168*a^3*d^6*D)/(1008*d^8*(b*c^2 + a*d^2)*(c + d*x)^6) - (b 
*(3430*b^3*c^6*C*d - 1180*b^3*B*c^5*d^2 + 250*A*b^3*c^4*d^3 + 7275*a*b^2*c 
^4*C*d^3 - 2352*a*b^2*B*c^3*d^4 + 501*a*A*b^2*c^2*d^5 + 4266*a^2*b*c^2*C*d 
^5 - 1161*a^2*b*B*c*d^6 + 240*a^2*A*b*d^7 + 432*a^3*C*d^7 - 7840*b^3*c^7*D 
 - 17790*a*b^2*c^5*d^2*D - 12075*a^2*b*c^3*d^4*D - 2136*a^3*c*d^6*D))/(100 
8*d^8*(b*c^2 + a*d^2)^2*(c + d*x)^5) - (b*(-7448*b^4*c^7*C*d + 1328*b^4*B* 
c^6*d^2 - 8*A*b^4*c^5*d^3 - 22280*a*b^3*c^5*C*d^3 + 3992*a*b^3*B*c^4*d^4 - 
 32*a*A*b^3*c^3*d^5 - 22137*a^2*b^2*c^3*C*d^5 + 4002*a^2*b^2*B*c^2*d^6 - 1 
23*a^2*A*b^2*c*d^7 - 7206*a^3*b*c*C*d^7 + 1239*a^3*b*B*d^8 + 25088*b^4*c^8 
*D + 77240*a*b^3*c^6*d^2*D + 81240*a^2*b^2*c^4*d^4*D + 31173*a^3*b*c^2*d^6 
*D + 2184*a^4*d^8*D))/(4032*d^8*(b*c^2 + a*d^2)^3*(c + d*x)^4) - (b^2*(...
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 637, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2182, 25, 2182, 25, 27, 679, 486, 486, 486, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx\)

\(\Big \downarrow \) 2182

\(\displaystyle -\frac {\int -\frac {\left (b x^2+a\right )^{5/2} \left (9 \left (\frac {b c^2}{d}+a d\right ) D x^2+\left (a (9 C d-9 c D)+b \left (-\frac {7 D c^3}{d^2}+\frac {7 C c^2}{d}+2 B c-2 A d\right )\right ) x+9 \left (A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^9}dx}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (b x^2+a\right )^{5/2} \left (9 \left (\frac {b c^2}{d}+a d\right ) D x^2+\left (9 a (C d-c D)+b \left (-\frac {7 D c^3}{d^2}+\frac {7 C c^2}{d}+2 B c-2 A d\right )\right ) x+9 \left (A b c-a \left (-\frac {D c^2}{d}+C c-B d\right )\right )\right )}{(c+d x)^9}dx}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}-\frac {\int -\frac {\left (8 \left (A b \left (9 b c^2-2 a d^2\right )+a \left (9 a d (C d-2 c D)-b c \left (\frac {7 D c^2}{d}+2 C c-11 B d\right )\right )\right ) d^2+\left (72 a^2 D d^4+9 a b \left (13 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (56 D c^3+7 C d c^2+2 B d^2 c-11 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{d^2 (c+d x)^8}dx}{8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (8 \left (A b \left (9 b c^2-2 a d^2\right )+a \left (9 a d (C d-2 c D)-b c \left (\frac {7 D c^2}{d}+2 C c-11 B d\right )\right )\right ) d^2+\left (72 a^2 D d^4+9 a b \left (13 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (56 D c^3+7 C d c^2+2 B d^2 c-11 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{d^2 (c+d x)^8}dx}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (8 \left (A b \left (9 b c^2-2 a d^2\right )+a \left (9 a d (C d-2 c D)-b c \left (\frac {7 D c^2}{d}+2 C c-11 B d\right )\right )\right ) d^2+\left (72 a^2 D d^4+9 a b \left (13 D c^2+2 C d c-B d^2\right ) d^2+b^2 c \left (56 D c^3+7 C d c^2+2 B d^2 c-11 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}}{(c+d x)^8}dx}{8 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {\frac {9 d^2 \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (-8 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+10 c C d\right )+b^2 c^2 (c C-10 B d)\right )\right ) \int \frac {\left (b x^2+a\right )^{5/2}}{(c+d x)^7}dx}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (72 a^2 d^4 (C d-3 c D)-a b d^2 \left (16 A d^3-97 B c d^2+173 c^3 D+34 c^2 C d\right )-b^2 c^2 \left (-83 A d^3+2 B c d^2+56 c^3 D+7 c^2 C d\right )\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {\frac {9 d^2 \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (-8 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+10 c C d\right )+b^2 c^2 (c C-10 B d)\right )\right ) \left (\frac {5 a b \int \frac {\left (b x^2+a\right )^{3/2}}{(c+d x)^5}dx}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (72 a^2 d^4 (C d-3 c D)-a b d^2 \left (16 A d^3-97 B c d^2+173 c^3 D+34 c^2 C d\right )-b^2 c^2 \left (-83 A d^3+2 B c d^2+56 c^3 D+7 c^2 C d\right )\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {\frac {9 d^2 \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (-8 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+10 c C d\right )+b^2 c^2 (c C-10 B d)\right )\right ) \left (\frac {5 a b \left (\frac {3 a b \int \frac {\sqrt {b x^2+a}}{(c+d x)^3}dx}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (72 a^2 d^4 (C d-3 c D)-a b d^2 \left (16 A d^3-97 B c d^2+173 c^3 D+34 c^2 C d\right )-b^2 c^2 \left (-83 A d^3+2 B c d^2+56 c^3 D+7 c^2 C d\right )\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {\frac {9 d^2 \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (-8 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+10 c C d\right )+b^2 c^2 (c C-10 B d)\right )\right ) \left (\frac {5 a b \left (\frac {3 a b \left (\frac {a b \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (72 a^2 d^4 (C d-3 c D)-a b d^2 \left (16 A d^3-97 B c d^2+173 c^3 D+34 c^2 C d\right )-b^2 c^2 \left (-83 A d^3+2 B c d^2+56 c^3 D+7 c^2 C d\right )\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {9 d^2 \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (-8 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+10 c C d\right )+b^2 c^2 (c C-10 B d)\right )\right ) \left (\frac {5 a b \left (\frac {3 a b \left (-\frac {a b \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (72 a^2 d^4 (C d-3 c D)-a b d^2 \left (16 A d^3-97 B c d^2+173 c^3 D+34 c^2 C d\right )-b^2 c^2 \left (-83 A d^3+2 B c d^2+56 c^3 D+7 c^2 C d\right )\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {9 d^2 \left (\frac {5 a b \left (\frac {3 a b \left (-\frac {a b \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 \left (a d^2+b c^2\right )^{3/2}}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right ) \left (A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (-8 a^2 d^3 D-a b d \left (-B d^2-3 c^2 D+10 c C d\right )+b^2 c^2 (c C-10 B d)\right )\right )}{a d^2+b c^2}-\frac {\left (a+b x^2\right )^{7/2} \left (72 a^2 d^4 (C d-3 c D)-a b d^2 \left (16 A d^3-97 B c d^2+173 c^3 D+34 c^2 C d\right )-b^2 c^2 \left (-83 A d^3+2 B c d^2+56 c^3 D+7 c^2 C d\right )\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}}{8 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (9 a d^2 \left (-B d^2-3 c^2 D+2 c C d\right )+b c \left (-11 A d^3+2 B c d^2-16 c^3 D+7 c^2 C d\right )\right )}{8 d^2 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{7/2} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{9 d^2 (c+d x)^9 \left (a d^2+b c^2\right )}\)

Input:

Int[((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^10,x]
 

Output:

-1/9*((c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(a + b*x^2)^(7/2))/(d^2*(b*c^2 + 
 a*d^2)*(c + d*x)^9) + (((9*a*d^2*(2*c*C*d - B*d^2 - 3*c^2*D) + b*c*(7*c^2 
*C*d + 2*B*c*d^2 - 11*A*d^3 - 16*c^3*D))*(a + b*x^2)^(7/2))/(8*d^2*(b*c^2 
+ a*d^2)*(c + d*x)^8) + (-1/7*((72*a^2*d^4*(C*d - 3*c*D) - b^2*c^2*(7*c^2* 
C*d + 2*B*c*d^2 - 83*A*d^3 + 56*c^3*D) - a*b*d^2*(34*c^2*C*d - 97*B*c*d^2 
+ 16*A*d^3 + 173*c^3*D))*(a + b*x^2)^(7/2))/((b*c^2 + a*d^2)*(c + d*x)^7) 
+ (9*d^2*(A*b^2*c*(8*b*c^2 - 3*a*d^2) - a*(b^2*c^2*(c*C - 10*B*d) - 8*a^2* 
d^3*D - a*b*d*(10*c*C*d - B*d^2 - 3*c^2*D)))*(-1/6*((a*d - b*c*x)*(a + b*x 
^2)^(5/2))/((b*c^2 + a*d^2)*(c + d*x)^6) + (5*a*b*(-1/4*((a*d - b*c*x)*(a 
+ b*x^2)^(3/2))/((b*c^2 + a*d^2)*(c + d*x)^4) + (3*a*b*(-1/2*((a*d - b*c*x 
)*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)*(c + d*x)^2) - (a*b*ArcTanh[(a*d - b*c 
*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(2*(b*c^2 + a*d^2)^(3/2))))/(4 
*(b*c^2 + a*d^2))))/(6*(b*c^2 + a*d^2))))/(b*c^2 + a*d^2))/(8*d^2*(b*c^2 + 
 a*d^2)))/(9*(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 486

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), 
x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2)))   Int[(c + d*x)^(n + 2)*(a + 
b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && 
GtQ[p, 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 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 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. \(111785\) vs. \(2(735)=1470\).

Time = 3.61 (sec) , antiderivative size = 111786, normalized size of antiderivative = 145.74

method	result	size
default	\(\text {Expression too large to display}\)	\(111786\)

Input:

int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x,method=_RETURNVERBOSE 
)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x, algorithm="fri 
cas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(5/2)*(D*x**3+C*x**2+B*x+A)/(d*x+c)**10,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57646 vs. \(2 (730) = 1460\).

Time = 2.92 (sec) , antiderivative size = 57646, normalized size of antiderivative = 75.16 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x, algorithm="max 
ima")
 

Output:

55/256*D*b^11*c^15*arcsinh(b*x/sqrt(a*b))/(b^(17/2)*c^16*d^9 + 8*a*b^(15/2 
)*c^14*d^11 + 28*a^2*b^(13/2)*c^12*d^13 + 56*a^3*b^(11/2)*c^10*d^15 + 70*a 
^4*b^(9/2)*c^8*d^17 + 56*a^5*b^(7/2)*c^6*d^19 + 28*a^6*b^(5/2)*c^4*d^21 + 
8*a^7*b^(3/2)*c^2*d^23 + a^8*sqrt(b)*d^25) - 55/256*C*b^11*c^14*arcsinh(b* 
x/sqrt(a*b))/(b^(17/2)*c^16*d^8 + 8*a*b^(15/2)*c^14*d^10 + 28*a^2*b^(13/2) 
*c^12*d^12 + 56*a^3*b^(11/2)*c^10*d^14 + 70*a^4*b^(9/2)*c^8*d^16 + 56*a^5* 
b^(7/2)*c^6*d^18 + 28*a^6*b^(5/2)*c^4*d^20 + 8*a^7*b^(3/2)*c^2*d^22 + a^8* 
sqrt(b)*d^24) + 55/256*D*a*b^10*c^13*arcsinh(b*x/sqrt(a*b))/(b^(17/2)*c^16 
*d^7 + 8*a*b^(15/2)*c^14*d^9 + 28*a^2*b^(13/2)*c^12*d^11 + 56*a^3*b^(11/2) 
*c^10*d^13 + 70*a^4*b^(9/2)*c^8*d^15 + 56*a^5*b^(7/2)*c^6*d^17 + 28*a^6*b^ 
(5/2)*c^4*d^19 + 8*a^7*b^(3/2)*c^2*d^21 + a^8*sqrt(b)*d^23) + 55/256*B*b^1 
1*c^13*arcsinh(b*x/sqrt(a*b))/(b^(17/2)*c^16*d^7 + 8*a*b^(15/2)*c^14*d^9 + 
 28*a^2*b^(13/2)*c^12*d^11 + 56*a^3*b^(11/2)*c^10*d^13 + 70*a^4*b^(9/2)*c^ 
8*d^15 + 56*a^5*b^(7/2)*c^6*d^17 + 28*a^6*b^(5/2)*c^4*d^19 + 8*a^7*b^(3/2) 
*c^2*d^21 + a^8*sqrt(b)*d^23) - 55/256*sqrt(b*x^2 + a)*D*b^10*c^13*x/(b^8* 
c^16*d^7 + 8*a*b^7*c^14*d^9 + 28*a^2*b^6*c^12*d^11 + 56*a^3*b^5*c^10*d^13 
+ 70*a^4*b^4*c^8*d^15 + 56*a^5*b^3*c^6*d^17 + 28*a^6*b^2*c^4*d^19 + 8*a^7* 
b*c^2*d^21 + a^8*d^23) - 55/256*C*a*b^10*c^12*arcsinh(b*x/sqrt(a*b))/(b^(1 
7/2)*c^16*d^6 + 8*a*b^(15/2)*c^14*d^8 + 28*a^2*b^(13/2)*c^12*d^10 + 56*a^3 
*b^(11/2)*c^10*d^12 + 70*a^4*b^(9/2)*c^8*d^14 + 56*a^5*b^(7/2)*c^6*d^16...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17669 vs. \(2 (730) = 1460\).

Time = 1.47 (sec) , antiderivative size = 17669, normalized size of antiderivative = 23.04 \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x, algorithm="gia 
c")
 

Output:

-5/64*(C*a^4*b^5*c^3 - 8*A*a^3*b^6*c^3 + 3*D*a^5*b^4*c^2*d - 10*B*a^4*b^5* 
c^2*d - 10*C*a^5*b^4*c*d^2 + 3*A*a^4*b^5*c*d^2 - 8*D*a^6*b^3*d^3 + B*a^5*b 
^4*d^3)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 
- a*d^2))/((b^6*c^12 + 6*a*b^5*c^10*d^2 + 15*a^2*b^4*c^8*d^4 + 20*a^3*b^3* 
c^6*d^6 + 15*a^4*b^2*c^4*d^8 + 6*a^5*b*c^2*d^10 + a^6*d^12)*sqrt(-b*c^2 - 
a*d^2)) + 1/4032*(8064*(sqrt(b)*x - sqrt(b*x^2 + a))^17*D*b^9*c^12*d^8 + 4 
8384*(sqrt(b)*x - sqrt(b*x^2 + a))^17*D*a*b^8*c^10*d^10 + 120960*(sqrt(b)* 
x - sqrt(b*x^2 + a))^17*D*a^2*b^7*c^8*d^12 + 161280*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^17*D*a^3*b^6*c^6*d^14 + 120960*(sqrt(b)*x - sqrt(b*x^2 + a))^17*D* 
a^4*b^5*c^4*d^16 + 315*(sqrt(b)*x - sqrt(b*x^2 + a))^17*C*a^4*b^5*c^3*d^17 
 - 2520*(sqrt(b)*x - sqrt(b*x^2 + a))^17*A*a^3*b^6*c^3*d^17 + 49329*(sqrt( 
b)*x - sqrt(b*x^2 + a))^17*D*a^5*b^4*c^2*d^18 - 3150*(sqrt(b)*x - sqrt(b*x 
^2 + a))^17*B*a^4*b^5*c^2*d^18 - 3150*(sqrt(b)*x - sqrt(b*x^2 + a))^17*C*a 
^5*b^4*c*d^19 + 945*(sqrt(b)*x - sqrt(b*x^2 + a))^17*A*a^4*b^5*c*d^19 + 55 
44*(sqrt(b)*x - sqrt(b*x^2 + a))^17*D*a^6*b^3*d^20 + 315*(sqrt(b)*x - sqrt 
(b*x^2 + a))^17*B*a^5*b^4*d^20 + 64512*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D* 
b^(19/2)*c^13*d^7 + 8064*(sqrt(b)*x - sqrt(b*x^2 + a))^16*C*b^(19/2)*c^12* 
d^8 + 387072*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D*a*b^(17/2)*c^11*d^9 + 4838 
4*(sqrt(b)*x - sqrt(b*x^2 + a))^16*C*a*b^(17/2)*c^10*d^10 + 967680*(sqrt(b 
)*x - sqrt(b*x^2 + a))^16*D*a^2*b^(15/2)*c^9*d^11 + 120960*(sqrt(b)*x -...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (c+d\,x\right )}^{10}} \,d x \] Input:

int(((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^10,x)
 

Output:

int(((a + b*x^2)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(c + d*x)^10, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{10}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}} \left (D x^{3}+C \,x^{2}+B x +A \right )}{\left (d x +c \right )^{10}}d x \] Input:

int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x)
 

Output:

int((b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A)/(d*x+c)^10,x)