\(\int (c+d x)^2 (a+b x^2)^{5/2} (A+B x+C x^2+D x^3) \, dx\) [86]

Optimal result

Integrand size = 34, antiderivative size = 485 \[ \int (c+d x)^2 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a^2 \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (c C+2 B d)-3 a d (C d+2 c D))\right ) x \sqrt {a+b x^2}}{256 b^2}+\frac {a \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (c C+2 B d)-3 a d (C d+2 c D))\right ) x \left (a+b x^2\right )^{3/2}}{384 b^2}+\frac {\left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (c C+2 B d)-3 a d (C d+2 c D))\right ) x \left (a+b x^2\right )^{5/2}}{480 b^2}+\frac {\left (b^2 c (B c+2 A d)+a^2 d^2 D-a b \left (2 c C d+B d^2+c^2 D\right )\right ) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {\left (10 b \left (c^2 C+2 B c d+A d^2\right )-3 a d (C d+2 c D)\right ) x \left (a+b x^2\right )^{7/2}}{80 b^2}+\frac {d (C d+2 c D) x^3 \left (a+b x^2\right )^{7/2}}{10 b}-\frac {\left (2 a d^2 D-b \left (2 c C d+B d^2+c^2 D\right )\right ) \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac {d^2 D \left (a+b x^2\right )^{11/2}}{11 b^3}+\frac {a^3 \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (c C+2 B d)-3 a d (C d+2 c D))\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{5/2}} \] Output:

1/256*a^2*(10*A*b*(-a*d^2+8*b*c^2)-a*(10*b*c*(2*B*d+C*c)-3*a*d*(C*d+2*D*c) 
))*x*(b*x^2+a)^(1/2)/b^2+1/384*a*(10*A*b*(-a*d^2+8*b*c^2)-a*(10*b*c*(2*B*d 
+C*c)-3*a*d*(C*d+2*D*c)))*x*(b*x^2+a)^(3/2)/b^2+1/480*(10*A*b*(-a*d^2+8*b* 
c^2)-a*(10*b*c*(2*B*d+C*c)-3*a*d*(C*d+2*D*c)))*x*(b*x^2+a)^(5/2)/b^2+1/7*( 
b^2*c*(2*A*d+B*c)+a^2*d^2*D-a*b*(B*d^2+2*C*c*d+D*c^2))*(b*x^2+a)^(7/2)/b^3 
+1/80*(10*b*(A*d^2+2*B*c*d+C*c^2)-3*a*d*(C*d+2*D*c))*x*(b*x^2+a)^(7/2)/b^2 
+1/10*d*(C*d+2*D*c)*x^3*(b*x^2+a)^(7/2)/b-1/9*(2*a*d^2*D-b*(B*d^2+2*C*c*d+ 
D*c^2))*(b*x^2+a)^(9/2)/b^3+1/11*d^2*D*(b*x^2+a)^(11/2)/b^3+1/256*a^3*(10* 
A*b*(-a*d^2+8*b*c^2)-a*(10*b*c*(2*B*d+C*c)-3*a*d*(C*d+2*D*c)))*arctanh(b^( 
1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 4.75 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.12 \[ \int (c+d x)^2 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {a+b x^2} \left (10240 a^5 d^2 D+10 a^3 b^2 \left (99 A d (256 c+35 d x)+22 B \left (576 c^2+315 c d x+64 d^2 x^2\right )+x \left (22 c d x (128 C+63 D x)+3 d^2 x^2 (231 C+128 D x)+11 c^2 (315 C+128 D x)\right )\right )-5 a^4 b \left (5632 c^2 D+22 c d (512 C+189 D x)+d^2 (5632 B+x (2079 C+1024 D x))\right )+32 b^5 x^5 \left (165 A \left (28 c^2+48 c d x+21 d^2 x^2\right )+x \left (110 B \left (36 c^2+63 c d x+28 d^2 x^2\right )+7 x \left (55 c^2 (9 C+8 D x)+88 c d x (10 C+9 D x)+36 d^2 x^2 (11 C+10 D x)\right )\right )\right )+16 a b^4 x^3 \left (165 A \left (182 c^2+288 c d x+119 d^2 x^2\right )+x \left (110 B \left (216 c^2+357 c d x+152 d^2 x^2\right )+x \left (55 c^2 (357 C+304 D x)+22 c d x (1520 C+1323 D x)+7 d^2 x^2 (2079 C+1840 D x)\right )\right )\right )+4 a^2 b^3 x \left (165 A \left (924 c^2+1152 c d x+413 d^2 x^2\right )+x \left (330 B \left (288 c^2+413 c d x+160 d^2 x^2\right )+x \left (165 c^2 (413 C+320 D x)+132 c d x (800 C+651 D x)+2 d^2 x^2 (21483 C+18080 D x)\right )\right )\right )\right )-3465 a^3 \sqrt {b} \left (10 A b \left (8 b c^2-a d^2\right )+a (-10 b c (c C+2 B d)+3 a d (C d+2 c D))\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{887040 b^3} \] Input:

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

Output:

(Sqrt[a + b*x^2]*(10240*a^5*d^2*D + 10*a^3*b^2*(99*A*d*(256*c + 35*d*x) + 
22*B*(576*c^2 + 315*c*d*x + 64*d^2*x^2) + x*(22*c*d*x*(128*C + 63*D*x) + 3 
*d^2*x^2*(231*C + 128*D*x) + 11*c^2*(315*C + 128*D*x))) - 5*a^4*b*(5632*c^ 
2*D + 22*c*d*(512*C + 189*D*x) + d^2*(5632*B + x*(2079*C + 1024*D*x))) + 3 
2*b^5*x^5*(165*A*(28*c^2 + 48*c*d*x + 21*d^2*x^2) + x*(110*B*(36*c^2 + 63* 
c*d*x + 28*d^2*x^2) + 7*x*(55*c^2*(9*C + 8*D*x) + 88*c*d*x*(10*C + 9*D*x) 
+ 36*d^2*x^2*(11*C + 10*D*x)))) + 16*a*b^4*x^3*(165*A*(182*c^2 + 288*c*d*x 
 + 119*d^2*x^2) + x*(110*B*(216*c^2 + 357*c*d*x + 152*d^2*x^2) + x*(55*c^2 
*(357*C + 304*D*x) + 22*c*d*x*(1520*C + 1323*D*x) + 7*d^2*x^2*(2079*C + 18 
40*D*x)))) + 4*a^2*b^3*x*(165*A*(924*c^2 + 1152*c*d*x + 413*d^2*x^2) + x*( 
330*B*(288*c^2 + 413*c*d*x + 160*d^2*x^2) + x*(165*c^2*(413*C + 320*D*x) + 
 132*c*d*x*(800*C + 651*D*x) + 2*d^2*x^2*(21483*C + 18080*D*x))))) - 3465* 
a^3*Sqrt[b]*(10*A*b*(8*b*c^2 - a*d^2) + a*(-10*b*c*(c*C + 2*B*d) + 3*a*d*( 
C*d + 2*c*D)))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(887040*b^3)
 

Rubi [A] (verified)

Time = 0.97 (sec) , antiderivative size = 431, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {2185, 2185, 27, 687, 676, 211, 211, 211, 224, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int (c+d x)^2 \left (b x^2+a\right )^{5/2} \left (b (11 C d-18 c D) x^2 d^2+(11 A b d-4 a c D) d^2+\left (-7 b D c^2+11 b B d^2-4 a d^2 D\right ) x d\right )dx}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\int b d^3 (c+d x)^2 \left (d (110 A b d-33 a C d+14 a c D)-\left (40 a D d^2+b \left (-56 D c^2+77 C d c-110 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{10 b d^2}+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{10} d \int (c+d x)^2 \left (d (110 A b d-33 a C d+14 a c D)-\left (40 a D d^2+b \left (-56 D c^2+77 C d c-110 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {1}{10} d \left (\frac {\int (c+d x) \left (d \left (990 A c d b^2+a \left (80 a d^2 D-b \left (-14 D c^2+143 C d c+220 B d^2\right )\right )\right )-b \left (a (297 C d-46 c D) d^2+2 b \left (-56 D c^3+77 C d c^2-110 B d^2 c-495 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{9 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^2 \left (40 a d^2 D+b \left (-110 B d^2-56 c^2 D+77 c C d\right )\right )}{9 b}\right )+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{10} d \left (\frac {\frac {99}{8} d^2 \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \int \left (b x^2+a\right )^{5/2}dx+\frac {2 \left (a+b x^2\right )^{7/2} \left (40 a^2 d^4 D-10 a b d^2 \left (11 B d^2-3 c^2 D+22 c C d\right )-b^2 c \left (-990 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{7 b}-\frac {1}{8} d x \left (a+b x^2\right )^{7/2} \left (a d^2 (297 C d-46 c D)+2 b \left (-495 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{9 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^2 \left (40 a d^2 D+b \left (-110 B d^2-56 c^2 D+77 c C d\right )\right )}{9 b}\right )+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{10} d \left (\frac {\frac {99}{8} d^2 \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {2 \left (a+b x^2\right )^{7/2} \left (40 a^2 d^4 D-10 a b d^2 \left (11 B d^2-3 c^2 D+22 c C d\right )-b^2 c \left (-990 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{7 b}-\frac {1}{8} d x \left (a+b x^2\right )^{7/2} \left (a d^2 (297 C d-46 c D)+2 b \left (-495 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{9 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^2 \left (40 a d^2 D+b \left (-110 B d^2-56 c^2 D+77 c C d\right )\right )}{9 b}\right )+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{10} d \left (\frac {\frac {99}{8} d^2 \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {2 \left (a+b x^2\right )^{7/2} \left (40 a^2 d^4 D-10 a b d^2 \left (11 B d^2-3 c^2 D+22 c C d\right )-b^2 c \left (-990 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{7 b}-\frac {1}{8} d x \left (a+b x^2\right )^{7/2} \left (a d^2 (297 C d-46 c D)+2 b \left (-495 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{9 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^2 \left (40 a d^2 D+b \left (-110 B d^2-56 c^2 D+77 c C d\right )\right )}{9 b}\right )+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{10} d \left (\frac {\frac {99}{8} d^2 \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {2 \left (a+b x^2\right )^{7/2} \left (40 a^2 d^4 D-10 a b d^2 \left (11 B d^2-3 c^2 D+22 c C d\right )-b^2 c \left (-990 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{7 b}-\frac {1}{8} d x \left (a+b x^2\right )^{7/2} \left (a d^2 (297 C d-46 c D)+2 b \left (-495 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{9 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^2 \left (40 a d^2 D+b \left (-110 B d^2-56 c^2 D+77 c C d\right )\right )}{9 b}\right )+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{10} d \left (\frac {\frac {99}{8} d^2 \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )+\frac {2 \left (a+b x^2\right )^{7/2} \left (40 a^2 d^4 D-10 a b d^2 \left (11 B d^2-3 c^2 D+22 c C d\right )-b^2 c \left (-990 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{7 b}-\frac {1}{8} d x \left (a+b x^2\right )^{7/2} \left (a d^2 (297 C d-46 c D)+2 b \left (-495 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{9 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^2 \left (40 a d^2 D+b \left (-110 B d^2-56 c^2 D+77 c C d\right )\right )}{9 b}\right )+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{10} d \left (\frac {\frac {2 \left (a+b x^2\right )^{7/2} \left (40 a^2 d^4 D-10 a b d^2 \left (11 B d^2-3 c^2 D+22 c C d\right )-b^2 c \left (-990 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{7 b}+\frac {99}{8} d^2 \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right ) \left (10 A b \left (8 b c^2-a d^2\right )-a (10 b c (2 B d+c C)-3 a d (2 c D+C d))\right )-\frac {1}{8} d x \left (a+b x^2\right )^{7/2} \left (a d^2 (297 C d-46 c D)+2 b \left (-495 A d^3-110 B c d^2-56 c^3 D+77 c^2 C d\right )\right )}{9 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^2 \left (40 a d^2 D+b \left (-110 B d^2-56 c^2 D+77 c C d\right )\right )}{9 b}\right )+\frac {1}{10} d \left (a+b x^2\right )^{7/2} (c+d x)^3 (11 C d-18 c D)}{11 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^4}{11 b d^2}\)

Input:

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

Output:

(D*(c + d*x)^4*(a + b*x^2)^(7/2))/(11*b*d^2) + ((d*(11*C*d - 18*c*D)*(c + 
d*x)^3*(a + b*x^2)^(7/2))/10 + (d*(-1/9*((40*a*d^2*D + b*(77*c*C*d - 110*B 
*d^2 - 56*c^2*D))*(c + d*x)^2*(a + b*x^2)^(7/2))/b + ((2*(40*a^2*d^4*D - 1 
0*a*b*d^2*(22*c*C*d + 11*B*d^2 - 3*c^2*D) - b^2*c*(77*c^2*C*d - 110*B*c*d^ 
2 - 990*A*d^3 - 56*c^3*D))*(a + b*x^2)^(7/2))/(7*b) - (d*(a*d^2*(297*C*d - 
 46*c*D) + 2*b*(77*c^2*C*d - 110*B*c*d^2 - 495*A*d^3 - 56*c^3*D))*x*(a + b 
*x^2)^(7/2))/8 + (99*d^2*(10*A*b*(8*b*c^2 - a*d^2) - a*(10*b*c*(c*C + 2*B* 
d) - 3*a*d*(C*d + 2*c*D)))*((x*(a + b*x^2)^(5/2))/6 + (5*a*((x*(a + b*x^2) 
^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + 
b*x^2]])/(2*Sqrt[b])))/4))/6))/8)/(9*b)))/10)/(11*b*d^3)
 

Defintions of rubi rules used

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -1]
 

rule 687

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])
 

rule 2185

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]))
 

Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 438, normalized size of antiderivative = 0.90

method	result	size
default	\(A \,c^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )+\frac {c \left (2 A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b}+d \left (C d +2 D c \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )+\left (B \,d^{2}+2 C c d +D c^{2}\right ) \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )+D d^{2} \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )}{11 b}\right )\)	\(438\)

Input:

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

Output:

A*c^2*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(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)))))+1/7*c*(2*A*d+B 
*c)*(b*x^2+a)^(7/2)/b+d*(C*d+2*D*c)*(1/10*x^3*(b*x^2+a)^(7/2)/b-3/10*a/b*( 
1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(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)))))))+(A*d^2+2*B*c*d+C*c^2)*(1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6* 
x*(b*x^2+a)^(5/2)+5/6*a*(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))))))+(B*d^2+2*C*c*d+D*c^2)*(1 
/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a/b^2*(b*x^2+a)^(7/2))+D*d^2*(1/11*x^4*(b*x^ 
2+a)^(7/2)/b-4/11*a/b*(1/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a/b^2*(b*x^2+a)^(7/2 
)))
 

Fricas [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1425, normalized size of antiderivative = 2.94 \[ \int (c+d x)^2 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

[1/1774080*(3465*(10*(C*a^4*b - 8*A*a^3*b^2)*c^2 - 2*(3*D*a^5 - 10*B*a^4*b 
)*c*d - (3*C*a^5 - 10*A*a^4*b)*d^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + 
a)*sqrt(b)*x - a) + 2*(80640*D*b^5*d^2*x^10 + 88704*(2*D*b^5*c*d + C*b^5*d 
^2)*x^9 + 8960*(11*D*b^5*c^2 + 22*C*b^5*c*d + (23*D*a*b^4 + 11*B*b^5)*d^2) 
*x^8 + 11088*(10*C*b^5*c^2 + 2*(21*D*a*b^4 + 10*B*b^5)*c*d + (21*C*a*b^4 + 
 10*A*b^5)*d^2)*x^7 + 1280*(11*(19*D*a*b^4 + 9*B*b^5)*c^2 + 22*(19*C*a*b^4 
 + 9*A*b^5)*c*d + (113*D*a^2*b^3 + 209*B*a*b^4)*d^2)*x^6 + 1848*(10*(17*C* 
a*b^4 + 8*A*b^5)*c^2 + 2*(93*D*a^2*b^3 + 170*B*a*b^4)*c*d + (93*C*a^2*b^3 
+ 170*A*a*b^4)*d^2)*x^5 + 3840*(11*(5*D*a^2*b^3 + 9*B*a*b^4)*c^2 + 22*(5*C 
*a^2*b^3 + 9*A*a*b^4)*c*d + (D*a^3*b^2 + 55*B*a^2*b^3)*d^2)*x^4 + 2310*(2* 
(59*C*a^2*b^3 + 104*A*a*b^4)*c^2 + 2*(3*D*a^3*b^2 + 118*B*a^2*b^3)*c*d + ( 
3*C*a^3*b^2 + 118*A*a^2*b^3)*d^2)*x^3 - 14080*(2*D*a^4*b - 9*B*a^3*b^2)*c^ 
2 - 28160*(2*C*a^4*b - 9*A*a^3*b^2)*c*d + 2560*(4*D*a^5 - 11*B*a^4*b)*d^2 
+ 1280*(11*(D*a^3*b^2 + 27*B*a^2*b^3)*c^2 + 22*(C*a^3*b^2 + 27*A*a^2*b^3)* 
c*d - (4*D*a^4*b - 11*B*a^3*b^2)*d^2)*x^2 + 3465*(2*(5*C*a^3*b^2 + 88*A*a^ 
2*b^3)*c^2 - 2*(3*D*a^4*b - 10*B*a^3*b^2)*c*d - (3*C*a^4*b - 10*A*a^3*b^2) 
*d^2)*x)*sqrt(b*x^2 + a))/b^3, 1/887040*(3465*(10*(C*a^4*b - 8*A*a^3*b^2)* 
c^2 - 2*(3*D*a^5 - 10*B*a^4*b)*c*d - (3*C*a^5 - 10*A*a^4*b)*d^2)*sqrt(-b)* 
arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (80640*D*b^5*d^2*x^10 + 88704*(2*D*b^ 
5*c*d + C*b^5*d^2)*x^9 + 8960*(11*D*b^5*c^2 + 22*C*b^5*c*d + (23*D*a*b^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2113 vs. \(2 (471) = 942\).

Time = 1.06 (sec) , antiderivative size = 2113, normalized size of antiderivative = 4.36 \[ \int (c+d x)^2 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x**2)*(D*b**2*d**2*x**10/11 + x**9*(C*b**3*d**2 + 2* 
D*b**3*c*d)/(10*b) + x**8*(B*b**3*d**2 + 2*C*b**3*c*d + 23*D*a*b**2*d**2/1 
1 + D*b**3*c**2)/(9*b) + x**7*(A*b**3*d**2 + 2*B*b**3*c*d + 3*C*a*b**2*d** 
2 + C*b**3*c**2 + 6*D*a*b**2*c*d - 9*a*(C*b**3*d**2 + 2*D*b**3*c*d)/(10*b) 
)/(8*b) + x**6*(2*A*b**3*c*d + 3*B*a*b**2*d**2 + B*b**3*c**2 + 6*C*a*b**2* 
c*d + 3*D*a**2*b*d**2 + 3*D*a*b**2*c**2 - 8*a*(B*b**3*d**2 + 2*C*b**3*c*d 
+ 23*D*a*b**2*d**2/11 + D*b**3*c**2)/(9*b))/(7*b) + x**5*(3*A*a*b**2*d**2 
+ A*b**3*c**2 + 6*B*a*b**2*c*d + 3*C*a**2*b*d**2 + 3*C*a*b**2*c**2 + 6*D*a 
**2*b*c*d - 7*a*(A*b**3*d**2 + 2*B*b**3*c*d + 3*C*a*b**2*d**2 + C*b**3*c** 
2 + 6*D*a*b**2*c*d - 9*a*(C*b**3*d**2 + 2*D*b**3*c*d)/(10*b))/(8*b))/(6*b) 
 + x**4*(6*A*a*b**2*c*d + 3*B*a**2*b*d**2 + 3*B*a*b**2*c**2 + 6*C*a**2*b*c 
*d + D*a**3*d**2 + 3*D*a**2*b*c**2 - 6*a*(2*A*b**3*c*d + 3*B*a*b**2*d**2 + 
 B*b**3*c**2 + 6*C*a*b**2*c*d + 3*D*a**2*b*d**2 + 3*D*a*b**2*c**2 - 8*a*(B 
*b**3*d**2 + 2*C*b**3*c*d + 23*D*a*b**2*d**2/11 + D*b**3*c**2)/(9*b))/(7*b 
))/(5*b) + x**3*(3*A*a**2*b*d**2 + 3*A*a*b**2*c**2 + 6*B*a**2*b*c*d + C*a* 
*3*d**2 + 3*C*a**2*b*c**2 + 2*D*a**3*c*d - 5*a*(3*A*a*b**2*d**2 + A*b**3*c 
**2 + 6*B*a*b**2*c*d + 3*C*a**2*b*d**2 + 3*C*a*b**2*c**2 + 6*D*a**2*b*c*d 
- 7*a*(A*b**3*d**2 + 2*B*b**3*c*d + 3*C*a*b**2*d**2 + C*b**3*c**2 + 6*D*a* 
b**2*c*d - 9*a*(C*b**3*d**2 + 2*D*b**3*c*d)/(10*b))/(8*b))/(6*b))/(4*b) + 
x**2*(6*A*a**2*b*c*d + B*a**3*d**2 + 3*B*a**2*b*c**2 + 2*C*a**3*c*d + D...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.18 \[ \int (c+d x)^2 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/11*(b*x^2 + a)^(7/2)*D*d^2*x^4/b - 4/99*(b*x^2 + a)^(7/2)*D*a*d^2*x^2/b^ 
2 + 1/6*(b*x^2 + a)^(5/2)*A*c^2*x + 5/24*(b*x^2 + a)^(3/2)*A*a*c^2*x + 5/1 
6*sqrt(b*x^2 + a)*A*a^2*c^2*x + 1/10*(2*D*c*d + C*d^2)*(b*x^2 + a)^(7/2)*x 
^3/b + 5/16*A*a^3*c^2*arcsinh(b*x/sqrt(a*b))/sqrt(b) + 1/7*(b*x^2 + a)^(7/ 
2)*B*c^2/b + 2/7*(b*x^2 + a)^(7/2)*A*c*d/b + 8/693*(b*x^2 + a)^(7/2)*D*a^2 
*d^2/b^3 + 1/9*(D*c^2 + 2*C*c*d + B*d^2)*(b*x^2 + a)^(7/2)*x^2/b - 3/80*(2 
*D*c*d + C*d^2)*(b*x^2 + a)^(7/2)*a*x/b^2 + 1/160*(2*D*c*d + C*d^2)*(b*x^2 
 + a)^(5/2)*a^2*x/b^2 + 1/128*(2*D*c*d + C*d^2)*(b*x^2 + a)^(3/2)*a^3*x/b^ 
2 + 3/256*(2*D*c*d + C*d^2)*sqrt(b*x^2 + a)*a^4*x/b^2 + 1/8*(C*c^2 + 2*B*c 
*d + A*d^2)*(b*x^2 + a)^(7/2)*x/b - 1/48*(C*c^2 + 2*B*c*d + A*d^2)*(b*x^2 
+ a)^(5/2)*a*x/b - 5/192*(C*c^2 + 2*B*c*d + A*d^2)*(b*x^2 + a)^(3/2)*a^2*x 
/b - 5/128*(C*c^2 + 2*B*c*d + A*d^2)*sqrt(b*x^2 + a)*a^3*x/b + 3/256*(2*D* 
c*d + C*d^2)*a^5*arcsinh(b*x/sqrt(a*b))/b^(5/2) - 5/128*(C*c^2 + 2*B*c*d + 
 A*d^2)*a^4*arcsinh(b*x/sqrt(a*b))/b^(3/2) - 2/63*(D*c^2 + 2*C*c*d + B*d^2 
)*(b*x^2 + a)^(7/2)*a/b^2
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 757, normalized size of antiderivative = 1.56 \[ \int (c+d x)^2 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/887040*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*(9*(10*D*b^2*d^2*x + 11*(2*D*b^ 
11*c*d + C*b^11*d^2)/b^9)*x + 10*(11*D*b^11*c^2 + 22*C*b^11*c*d + 23*D*a*b 
^10*d^2 + 11*B*b^11*d^2)/b^9)*x + 99*(10*C*b^11*c^2 + 42*D*a*b^10*c*d + 20 
*B*b^11*c*d + 21*C*a*b^10*d^2 + 10*A*b^11*d^2)/b^9)*x + 80*(209*D*a*b^10*c 
^2 + 99*B*b^11*c^2 + 418*C*a*b^10*c*d + 198*A*b^11*c*d + 113*D*a^2*b^9*d^2 
 + 209*B*a*b^10*d^2)/b^9)*x + 231*(170*C*a*b^10*c^2 + 80*A*b^11*c^2 + 186* 
D*a^2*b^9*c*d + 340*B*a*b^10*c*d + 93*C*a^2*b^9*d^2 + 170*A*a*b^10*d^2)/b^ 
9)*x + 480*(55*D*a^2*b^9*c^2 + 99*B*a*b^10*c^2 + 110*C*a^2*b^9*c*d + 198*A 
*a*b^10*c*d + D*a^3*b^8*d^2 + 55*B*a^2*b^9*d^2)/b^9)*x + 1155*(118*C*a^2*b 
^9*c^2 + 208*A*a*b^10*c^2 + 6*D*a^3*b^8*c*d + 236*B*a^2*b^9*c*d + 3*C*a^3* 
b^8*d^2 + 118*A*a^2*b^9*d^2)/b^9)*x + 640*(11*D*a^3*b^8*c^2 + 297*B*a^2*b^ 
9*c^2 + 22*C*a^3*b^8*c*d + 594*A*a^2*b^9*c*d - 4*D*a^4*b^7*d^2 + 11*B*a^3* 
b^8*d^2)/b^9)*x + 3465*(10*C*a^3*b^8*c^2 + 176*A*a^2*b^9*c^2 - 6*D*a^4*b^7 
*c*d + 20*B*a^3*b^8*c*d - 3*C*a^4*b^7*d^2 + 10*A*a^3*b^8*d^2)/b^9)*x - 128 
0*(22*D*a^4*b^7*c^2 - 99*B*a^3*b^8*c^2 + 44*C*a^4*b^7*c*d - 198*A*a^3*b^8* 
c*d - 8*D*a^5*b^6*d^2 + 22*B*a^4*b^7*d^2)/b^9) + 1/256*(10*C*a^4*b*c^2 - 8 
0*A*a^3*b^2*c^2 - 6*D*a^5*c*d + 20*B*a^4*b*c*d - 3*C*a^5*d^2 + 10*A*a^4*b* 
d^2)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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