\(\int (a+b x)^{5/2} \sqrt {c+d x} (A+B x+C x^2+D x^3) \, dx\) [120]

Optimal result

Integrand size = 34, antiderivative size = 762 \[ \int (a+b x)^{5/2} \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {(b c-a d)^3 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (28 c C d-24 B d^2-27 c^2 D\right )-b^3 \left (42 c^2 C d-56 B c d^2+80 A d^3-33 c^3 D\right )\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^4 d^6}+\frac {(b c-a d)^2 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (28 c C d-24 B d^2-27 c^2 D\right )-b^3 \left (42 c^2 C d-56 B c d^2+80 A d^3-33 c^3 D\right )\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^4 d^5}-\frac {(b c-a d) \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (28 c C d-24 B d^2-27 c^2 D\right )-b^3 \left (42 c^2 C d-56 B c d^2+80 A d^3-33 c^3 D\right )\right ) (a+b x)^{5/2} \sqrt {c+d x}}{1920 b^4 d^4}-\frac {\left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (28 c C d-24 B d^2-27 c^2 D\right )-b^3 \left (42 c^2 C d-56 B c d^2+80 A d^3-33 c^3 D\right )\right ) (a+b x)^{7/2} \sqrt {c+d x}}{320 b^4 d^3}-\frac {\left (42 b c C-56 b B d+70 a C d-60 a c D-\frac {33 b c^2 D}{d}-\frac {75 a^2 d D}{b}\right ) (a+b x)^{7/2} (c+d x)^{3/2}}{280 b^2 d^2}+\frac {(14 b C d-11 b c D-31 a d D) (a+b x)^{9/2} (c+d x)^{3/2}}{84 b^3 d^2}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}+\frac {(b c-a d)^4 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (28 c C d-24 B d^2-27 c^2 D\right )-b^3 \left (42 c^2 C d-56 B c d^2+80 A d^3-33 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{9/2} d^{13/2}} \] Output:

-1/1024*(-a*d+b*c)^3*(5*a^3*d^3*D-5*a^2*b*d^2*(2*C*d-3*D*c)-a*b^2*d*(-24*B 
*d^2+28*C*c*d-27*D*c^2)-b^3*(80*A*d^3-56*B*c*d^2+42*C*c^2*d-33*D*c^3))*(b* 
x+a)^(1/2)*(d*x+c)^(1/2)/b^4/d^6+1/1536*(-a*d+b*c)^2*(5*a^3*d^3*D-5*a^2*b* 
d^2*(2*C*d-3*D*c)-a*b^2*d*(-24*B*d^2+28*C*c*d-27*D*c^2)-b^3*(80*A*d^3-56*B 
*c*d^2+42*C*c^2*d-33*D*c^3))*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^4/d^5-1/1920*(- 
a*d+b*c)*(5*a^3*d^3*D-5*a^2*b*d^2*(2*C*d-3*D*c)-a*b^2*d*(-24*B*d^2+28*C*c* 
d-27*D*c^2)-b^3*(80*A*d^3-56*B*c*d^2+42*C*c^2*d-33*D*c^3))*(b*x+a)^(5/2)*( 
d*x+c)^(1/2)/b^4/d^4-1/320*(5*a^3*d^3*D-5*a^2*b*d^2*(2*C*d-3*D*c)-a*b^2*d* 
(-24*B*d^2+28*C*c*d-27*D*c^2)-b^3*(80*A*d^3-56*B*c*d^2+42*C*c^2*d-33*D*c^3 
))*(b*x+a)^(7/2)*(d*x+c)^(1/2)/b^4/d^3-1/280*(42*C*b*c-56*B*b*d+70*C*a*d-6 
0*D*a*c-33*b*c^2*D/d-75*a^2*d*D/b)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^2/d^2+1/8 
4*(14*C*b*d-31*D*a*d-11*D*b*c)*(b*x+a)^(9/2)*(d*x+c)^(3/2)/b^3/d^2+1/7*D*( 
b*x+a)^(11/2)*(d*x+c)^(3/2)/b^3/d+1/1024*(-a*d+b*c)^4*(5*a^3*d^3*D-5*a^2*b 
*d^2*(2*C*d-3*D*c)-a*b^2*d*(-24*B*d^2+28*C*c*d-27*D*c^2)-b^3*(80*A*d^3-56* 
B*c*d^2+42*C*c^2*d-33*D*c^3))*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c 
)^(1/2))/b^(9/2)/d^(13/2)
 

Mathematica [A] (verified)

Time = 2.98 (sec) , antiderivative size = 726, normalized size of antiderivative = 0.95 \[ \int (a+b x)^{5/2} \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-525 a^6 d^6 D+350 a^5 b d^5 (3 C d+D (c+d x))-35 a^4 b^2 d^4 \left (-11 c^2 D+c (26 C d+6 d D x)+4 d^2 (18 B+x (5 C+2 D x))\right )+20 a^3 b^3 d^3 \left (33 c^3 D-c^2 d (63 C+13 D x)+4 c d^2 (42 B+x (7 C+2 D x))+4 d^3 \left (105 A+x \left (21 B+7 C x+3 D x^2\right )\right )\right )+2 a b^5 d \left (4935 c^5 D-189 c^4 d (35 C+17 D x)+8 c^3 d^2 (1190 B+x (539 C+318 D x))+32 c d^4 x \left (315 A+x \left (154 B+91 C x+60 D x^2\right )\right )-8 c^2 d^3 \left (1925 A+x \left (777 B+427 C x+271 D x^2\right )\right )+64 d^5 x^2 \left (595 A+x \left (441 B+350 C x+290 D x^2\right )\right )\right )+a^2 b^4 d^2 \left (-8043 c^4 D+4 c^3 d (2933 C+1251 D x)-8 c^2 d^2 (2422 B+x (917 C+486 D x))+16 c d^3 \left (2555 A+x \left (763 B+357 C x+205 D x^2\right )\right )+32 d^4 x \left (2065 A+x \left (1302 B+945 C x+740 D x^2\right )\right )\right )+b^6 \left (-3465 c^6 D+210 c^5 d (21 C+11 D x)-84 c^4 d^2 (70 B+x (35 C+22 D x))+128 c d^5 x^2 (35 A+x (21 B+2 x (7 C+5 D x)))+256 d^6 x^3 (105 A+2 x (42 B+5 x (7 C+6 D x)))+16 c^3 d^3 (525 A+x (245 B+3 x (49 C+33 D x)))-32 c^2 d^4 x (175 A+x (98 B+x (63 C+44 D x)))\right )\right )}{107520 b^4 d^6}+\frac {(b c-a d)^4 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)+a b^2 d \left (-28 c C d+24 B d^2+27 c^2 D\right )+b^3 \left (-42 c^2 C d+56 B c d^2-80 A d^3+33 c^3 D\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{1024 b^{9/2} d^{13/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-525*a^6*d^6*D + 350*a^5*b*d^5*(3*C*d + D*(c 
 + d*x)) - 35*a^4*b^2*d^4*(-11*c^2*D + c*(26*C*d + 6*d*D*x) + 4*d^2*(18*B 
+ x*(5*C + 2*D*x))) + 20*a^3*b^3*d^3*(33*c^3*D - c^2*d*(63*C + 13*D*x) + 4 
*c*d^2*(42*B + x*(7*C + 2*D*x)) + 4*d^3*(105*A + x*(21*B + 7*C*x + 3*D*x^2 
))) + 2*a*b^5*d*(4935*c^5*D - 189*c^4*d*(35*C + 17*D*x) + 8*c^3*d^2*(1190* 
B + x*(539*C + 318*D*x)) + 32*c*d^4*x*(315*A + x*(154*B + 91*C*x + 60*D*x^ 
2)) - 8*c^2*d^3*(1925*A + x*(777*B + 427*C*x + 271*D*x^2)) + 64*d^5*x^2*(5 
95*A + x*(441*B + 350*C*x + 290*D*x^2))) + a^2*b^4*d^2*(-8043*c^4*D + 4*c^ 
3*d*(2933*C + 1251*D*x) - 8*c^2*d^2*(2422*B + x*(917*C + 486*D*x)) + 16*c* 
d^3*(2555*A + x*(763*B + 357*C*x + 205*D*x^2)) + 32*d^4*x*(2065*A + x*(130 
2*B + 945*C*x + 740*D*x^2))) + b^6*(-3465*c^6*D + 210*c^5*d*(21*C + 11*D*x 
) - 84*c^4*d^2*(70*B + x*(35*C + 22*D*x)) + 128*c*d^5*x^2*(35*A + x*(21*B 
+ 2*x*(7*C + 5*D*x))) + 256*d^6*x^3*(105*A + 2*x*(42*B + 5*x*(7*C + 6*D*x) 
)) + 16*c^3*d^3*(525*A + x*(245*B + 3*x*(49*C + 33*D*x))) - 32*c^2*d^4*x*( 
175*A + x*(98*B + x*(63*C + 44*D*x))))))/(107520*b^4*d^6) + ((b*c - a*d)^4 
*(5*a^3*d^3*D - 5*a^2*b*d^2*(2*C*d - 3*c*D) + a*b^2*d*(-28*c*C*d + 24*B*d^ 
2 + 27*c^2*D) + b^3*(-42*c^2*C*d + 56*B*c*d^2 - 80*A*d^3 + 33*c^3*D))*ArcT 
anh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(1024*b^(9/2)*d^(13/ 
2))
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.59, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2125, 27, 1194, 27, 90, 60, 60, 60, 60, 66, 221}

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.

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

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\int \frac {1}{2} (a+b x)^{5/2} \sqrt {c+d x} \left (14 A d b^3+(14 b C d-31 a D d-11 b c D) x^2 b^2+2 \left (-10 d D a^2-11 b c D a+7 b^2 B d\right ) x b-a^2 (11 b c+3 a d) D\right )dx}{7 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (a+b x)^{5/2} \sqrt {c+d x} \left (14 A d b^3+(14 b C d-31 a D d-11 b c D) x^2 b^2+2 \left (-10 d D a^2-11 b c D a+7 b^2 B d\right ) x b-a^2 (11 b c+3 a d) D\right )dx}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 1194

\(\displaystyle \frac {\frac {\int \frac {3}{2} b^2 (a+b x)^{5/2} \sqrt {c+d x} \left (19 d^2 D a^3-2 b d (7 C d-30 c D) a^2-3 b^2 c (14 C d-11 c D) a+56 A b^3 d^2+b \left (-\left (\left (-33 D c^2+42 C d c-56 B d^2\right ) b^2\right )-10 a d (7 C d-6 c D) b+75 a^2 d^2 D\right ) x\right )dx}{6 b^2 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int (a+b x)^{5/2} \sqrt {c+d x} \left (19 d^2 D a^3-2 b d (7 C d-30 c D) a^2-3 b^2 c (14 C d-11 c D) a+56 A b^3 d^2+b \left (-\left (\left (-33 D c^2+42 C d c-56 B d^2\right ) b^2\right )-10 a d (7 C d-6 c D) b+75 a^2 d^2 D\right ) x\right )dx}{4 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {\frac {(a+b x)^{7/2} (c+d x)^{3/2} \left (75 a^2 d^2 D-10 a b d (7 C d-6 c D)-\left (b^2 \left (-56 B d^2-33 c^2 D+42 c C d\right )\right )\right )}{5 d}-\frac {7 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (-24 B d^2-27 c^2 D+28 c C d\right )-\left (b^3 \left (80 A d^3-56 B c d^2-33 c^3 D+42 c^2 C d\right )\right )\right ) \int (a+b x)^{5/2} \sqrt {c+d x}dx}{10 d}}{4 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {(a+b x)^{7/2} (c+d x)^{3/2} \left (75 a^2 d^2 D-10 a b d (7 C d-6 c D)-\left (b^2 \left (-56 B d^2-33 c^2 D+42 c C d\right )\right )\right )}{5 d}-\frac {7 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (-24 B d^2-27 c^2 D+28 c C d\right )-\left (b^3 \left (80 A d^3-56 B c d^2-33 c^3 D+42 c^2 C d\right )\right )\right ) \left (\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 d}}{4 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {(a+b x)^{7/2} (c+d x)^{3/2} \left (75 a^2 d^2 D-10 a b d (7 C d-6 c D)-\left (b^2 \left (-56 B d^2-33 c^2 D+42 c C d\right )\right )\right )}{5 d}-\frac {7 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (-24 B d^2-27 c^2 D+28 c C d\right )-\left (b^3 \left (80 A d^3-56 B c d^2-33 c^3 D+42 c^2 C d\right )\right )\right ) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 d}}{4 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {(a+b x)^{7/2} (c+d x)^{3/2} \left (75 a^2 d^2 D-10 a b d (7 C d-6 c D)-\left (b^2 \left (-56 B d^2-33 c^2 D+42 c C d\right )\right )\right )}{5 d}-\frac {7 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (-24 B d^2-27 c^2 D+28 c C d\right )-\left (b^3 \left (80 A d^3-56 B c d^2-33 c^3 D+42 c^2 C d\right )\right )\right ) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 d}}{4 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {\frac {(a+b x)^{7/2} (c+d x)^{3/2} \left (75 a^2 d^2 D-10 a b d (7 C d-6 c D)-\left (b^2 \left (-56 B d^2-33 c^2 D+42 c C d\right )\right )\right )}{5 d}-\frac {7 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (-24 B d^2-27 c^2 D+28 c C d\right )-\left (b^3 \left (80 A d^3-56 B c d^2-33 c^3 D+42 c^2 C d\right )\right )\right ) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 d}}{4 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\frac {\frac {(a+b x)^{7/2} (c+d x)^{3/2} \left (75 a^2 d^2 D-10 a b d (7 C d-6 c D)-\left (b^2 \left (-56 B d^2-33 c^2 D+42 c C d\right )\right )\right )}{5 d}-\frac {7 \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (-24 B d^2-27 c^2 D+28 c C d\right )-\left (b^3 \left (80 A d^3-56 B c d^2-33 c^3 D+42 c^2 C d\right )\right )\right ) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 d}}{4 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {(a+b x)^{7/2} (c+d x)^{3/2} \left (75 a^2 d^2 D-10 a b d (7 C d-6 c D)-\left (b^2 \left (-56 B d^2-33 c^2 D+42 c C d\right )\right )\right )}{5 d}-\frac {7 \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right ) \left (5 a^3 d^3 D-5 a^2 b d^2 (2 C d-3 c D)-a b^2 d \left (-24 B d^2-27 c^2 D+28 c C d\right )-\left (b^3 \left (80 A d^3-56 B c d^2-33 c^3 D+42 c^2 C d\right )\right )\right )}{10 d}}{4 d}+\frac {(a+b x)^{9/2} (c+d x)^{3/2} (-31 a d D-11 b c D+14 b C d)}{6 d}}{14 b^3 d}+\frac {D (a+b x)^{11/2} (c+d x)^{3/2}}{7 b^3 d}\)

Input:

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

Output:

(D*(a + b*x)^(11/2)*(c + d*x)^(3/2))/(7*b^3*d) + (((14*b*C*d - 11*b*c*D - 
31*a*d*D)*(a + b*x)^(9/2)*(c + d*x)^(3/2))/(6*d) + (((75*a^2*d^2*D - 10*a* 
b*d*(7*C*d - 6*c*D) - b^2*(42*c*C*d - 56*B*d^2 - 33*c^2*D))*(a + b*x)^(7/2 
)*(c + d*x)^(3/2))/(5*d) - (7*(5*a^3*d^3*D - 5*a^2*b*d^2*(2*C*d - 3*c*D) - 
 a*b^2*d*(28*c*C*d - 24*B*d^2 - 27*c^2*D) - b^3*(42*c^2*C*d - 56*B*c*d^2 + 
 80*A*d^3 - 33*c^3*D))*(((a + b*x)^(7/2)*Sqrt[c + d*x])/(4*b) + ((b*c - a* 
d)*(((a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d) - (5*(b*c - a*d)*(((a + b*x)^(3/ 
2)*Sqrt[c + d*x])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d 
- ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/( 
Sqrt[b]*d^(3/2))))/(4*d)))/(6*d)))/(8*b)))/(10*d))/(4*d))/(14*b^3*d)
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1194

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x 
)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 
*p + 1))   Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 
2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) 
*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ 
[p, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
 

rule 2125

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x 
)^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Simp[1/(d*b^q*( 
m + n + q + 1))   Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q 
+ 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x) 
^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, 
 x] && PolyQ[Px, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3757\) vs. \(2(706)=1412\).

Time = 0.52 (sec) , antiderivative size = 3758, normalized size of antiderivative = 4.93

method	result	size
default	\(\text {Expression too large to display}\)	\(3758\)

Input:

int((b*x+a)^(5/2)*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOS 
E)
 

Output:

-1/215040*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-38080*B*a*b^5*c^3*d^3*((b*x+a)*(d* 
x+c))^(1/2)*(d*b)^(1/2)-3360*B*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^3*b^3 
*d^6*x-7840*B*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^6*c^3*d^3*x+5040*B*((b 
*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^4*b^2*d^6+11760*B*((b*x+a)*(d*x+c))^(1/ 
2)*(d*b)^(1/2)*b^6*c^4*d^2-2100*C*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^5* 
b*d^6-8820*C*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*b^6*c^5*d-33600*A*ln(1/2* 
(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^3*b 
^4*c*d^6+50400*A*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d 
+b*c)/(d*b)^(1/2))*a^2*b^5*c^2*d^5-33600*A*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x 
+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^6*c^3*d^4+4200*B*ln(1/2*( 
2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^4*b^ 
3*c*d^6+8400*B*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b 
*c)/(d*b)^(1/2))*a^3*b^4*c^2*d^5-25200*B*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c 
))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*b^5*c^3*d^4+21000*B*ln(1/2* 
(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^6 
*c^4*d^3-1260*C*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+ 
b*c)/(d*b)^(1/2))*a^5*b^2*c*d^6-1050*C*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c)) 
^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^4*b^3*c^2*d^5-4200*C*ln(1/2*(2* 
b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^3*b^4* 
c^3*d^4+15750*C*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a...
 

Fricas [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 2320, normalized size of antiderivative = 3.04 \[ \int (a+b x)^{5/2} \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="fr 
icas")
 

Output:

[-1/430080*(105*(33*D*b^7*c^7 - 21*(5*D*a*b^6 + 2*C*b^7)*c^6*d + 7*(15*D*a 
^2*b^5 + 20*C*a*b^6 + 8*B*b^7)*c^5*d^2 - 5*(5*D*a^3*b^4 + 30*C*a^2*b^5 + 4 
0*B*a*b^6 + 16*A*b^7)*c^4*d^3 - 5*(D*a^4*b^3 - 8*C*a^3*b^4 - 48*B*a^2*b^5 
- 64*A*a*b^6)*c^3*d^4 - (3*D*a^5*b^2 - 10*C*a^4*b^3 + 80*B*a^3*b^4 + 480*A 
*a^2*b^5)*c^2*d^5 - (5*D*a^6*b - 12*C*a^5*b^2 + 40*B*a^4*b^3 - 320*A*a^3*b 
^4)*c*d^6 + (5*D*a^7 - 10*C*a^6*b + 24*B*a^5*b^2 - 80*A*a^4*b^3)*d^7)*sqrt 
(b*d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 - 4*(2*b*d*x + b*c 
 + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 
 4*(15360*D*b^7*d^7*x^6 - 3465*D*b^7*c^6*d + 210*(47*D*a*b^6 + 21*C*b^7)*c 
^5*d^2 - 21*(383*D*a^2*b^5 + 630*C*a*b^6 + 280*B*b^7)*c^4*d^3 + 4*(165*D*a 
^3*b^4 + 2933*C*a^2*b^5 + 4760*B*a*b^6 + 2100*A*b^7)*c^3*d^4 + 7*(55*D*a^4 
*b^3 - 180*C*a^3*b^4 - 2768*B*a^2*b^5 - 4400*A*a*b^6)*c^2*d^5 + 70*(5*D*a^ 
5*b^2 - 13*C*a^4*b^3 + 48*B*a^3*b^4 + 584*A*a^2*b^5)*c*d^6 - 105*(5*D*a^6* 
b - 10*C*a^5*b^2 + 24*B*a^4*b^3 - 80*A*a^3*b^4)*d^7 + 1280*(D*b^7*c*d^6 + 
(29*D*a*b^6 + 14*C*b^7)*d^7)*x^5 - 128*(11*D*b^7*c^2*d^5 - 2*(15*D*a*b^6 + 
 7*C*b^7)*c*d^6 - (185*D*a^2*b^5 + 350*C*a*b^6 + 168*B*b^7)*d^7)*x^4 + 16* 
(99*D*b^7*c^3*d^4 - (271*D*a*b^6 + 126*C*b^7)*c^2*d^5 + (205*D*a^2*b^5 + 3 
64*C*a*b^6 + 168*B*b^7)*c*d^6 + 3*(5*D*a^3*b^4 + 630*C*a^2*b^5 + 1176*B*a* 
b^6 + 560*A*b^7)*d^7)*x^3 - 8*(231*D*b^7*c^4*d^3 - 6*(106*D*a*b^6 + 49*C*b 
^7)*c^3*d^4 + 2*(243*D*a^2*b^5 + 427*C*a*b^6 + 196*B*b^7)*c^2*d^5 - 2*(...
 

Sympy [F]

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

integrate((b*x+a)**(5/2)*(d*x+c)**(1/2)*(D*x**3+C*x**2+B*x+A),x)
 

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int (a+b x)^{5/2} \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="ma 
xima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4652 vs. \(2 (696) = 1392\).

Time = 0.71 (sec) , antiderivative size = 4652, normalized size of antiderivative = 6.10 \[ \int (a+b x)^{5/2} \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="gi 
ac")
 

Output:

1/107520*(1680*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + 
 a)*(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13* 
c^2*d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3* 
d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))* 
sqrt(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3* 
b*c*d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x 
 + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*B*a*abs(b) + 42*(sqrt(b^2*c + (b 
*x + a)*b*d - a*b*d)*(2*(4*(2*(b*x + a)*(8*(b*x + a)*(10*(b*x + a)/b^5 + ( 
b^30*c*d^9 - 61*a*b^29*d^10)/(b^34*d^10)) - 3*(3*b^31*c^2*d^8 + 14*a*b^30* 
c*d^9 - 417*a^2*b^29*d^10)/(b^34*d^10)) + (21*b^32*c^3*d^7 + 77*a*b^31*c^2 
*d^8 + 183*a^2*b^30*c*d^9 - 3481*a^3*b^29*d^10)/(b^34*d^10))*(b*x + a) - 5 
*(21*b^33*c^4*d^6 + 56*a*b^32*c^3*d^7 + 106*a^2*b^31*c^2*d^8 + 176*a^3*b^3 
0*c*d^9 - 2279*a^4*b^29*d^10)/(b^34*d^10))*(b*x + a) + 15*(21*b^34*c^5*d^5 
 + 35*a*b^33*c^4*d^6 + 50*a^2*b^32*c^3*d^7 + 70*a^3*b^31*c^2*d^8 + 105*a^4 
*b^30*c*d^9 - 793*a^5*b^29*d^10)/(b^34*d^10))*sqrt(b*x + a) + 15*(21*b^6*c 
^6 + 14*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 + 20*a^3*b^3*c^3*d^3 + 35*a^4*b^2 
*c^2*d^4 + 126*a^5*b*c*d^5 - 231*a^6*d^6)*log(abs(-sqrt(b*d)*sqrt(b*x + a) 
 + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^4*d^5))*D*a*abs(b) - 
 107520*((b^2*c - a*b*d)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + ( 
b*x + a)*b*d - a*b*d)))/sqrt(b*d) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)...
 

Mupad [F(-1)]

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

int((a + b*x)^(5/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D),x)
                                                                                    
                                                                                    
 

Output:

int((a + b*x)^(5/2)*(c + d*x)^(1/2)*(A + B*x + C*x^2 + x^3*D), x)
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 1752, normalized size of antiderivative = 2.30 \[ \int (a+b x)^{5/2} \sqrt {c+d x} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(5/2)*(d*x+c)^(1/2)*(D*x^3+C*x^2+B*x+A),x)
 

Output:

( - 525*sqrt(c + d*x)*sqrt(a + b*x)*a**6*b*d**7 + 1400*sqrt(c + d*x)*sqrt( 
a + b*x)*a**5*b**2*c*d**6 + 350*sqrt(c + d*x)*sqrt(a + b*x)*a**5*b**2*d**7 
*x + 5880*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**4*d**6 - 525*sqrt(c + d*x)*s 
qrt(a + b*x)*a**4*b**3*c**2*d**5 - 910*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b* 
*3*c*d**6*x - 280*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b**3*d**7*x**2 + 44240* 
sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**5*c*d**5 + 67760*sqrt(c + d*x)*sqrt(a 
+ b*x)*a**3*b**5*d**6*x - 600*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**4*c**3*d 
**4 + 300*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**4*c**2*d**5*x + 720*sqrt(c + 
 d*x)*sqrt(a + b*x)*a**3*b**4*c*d**6*x**2 + 240*sqrt(c + d*x)*sqrt(a + b*x 
)*a**3*b**4*d**7*x**3 - 50176*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**6*c**2*d 
**4 + 32368*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**6*c*d**5*x + 117824*sqrt(c 
 + d*x)*sqrt(a + b*x)*a**2*b**6*d**6*x**2 + 3689*sqrt(c + d*x)*sqrt(a + b* 
x)*a**2*b**5*c**4*d**3 - 2332*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**5*c**3*d 
**4*x + 1824*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**5*c**2*d**5*x**2 + 33520* 
sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**5*c*d**6*x**3 + 23680*sqrt(c + d*x)*sq 
rt(a + b*x)*a**2*b**5*d**7*x**4 + 27440*sqrt(c + d*x)*sqrt(a + b*x)*a*b**7 
*c**3*d**3 - 18032*sqrt(c + d*x)*sqrt(a + b*x)*a*b**7*c**2*d**4*x + 14336* 
sqrt(c + d*x)*sqrt(a + b*x)*a*b**7*c*d**5*x**2 + 83328*sqrt(c + d*x)*sqrt( 
a + b*x)*a*b**7*d**6*x**3 - 3360*sqrt(c + d*x)*sqrt(a + b*x)*a*b**6*c**5*d 
**2 + 2198*sqrt(c + d*x)*sqrt(a + b*x)*a*b**6*c**4*d**3*x - 1744*sqrt(c...