\(\int (e x)^{3/2} (c+d x)^{5/2} (a+b x^2)^2 \, dx\) [811]

Optimal result

Integrand size = 26, antiderivative size = 483 \[ \int (e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=-\frac {c^4 \left (55 b^2 c^4+320 a b c^2 d^2+768 a^2 d^4\right ) e \sqrt {e x} \sqrt {c+d x}}{32768 d^6}+\frac {c^3 \left (55 b^2 c^4+320 a b c^2 d^2+768 a^2 d^4\right ) (e x)^{3/2} \sqrt {c+d x}}{49152 d^5}+\frac {31 c^2 \left (55 b^2 c^4+320 a b c^2 d^2+768 a^2 d^4\right ) (e x)^{5/2} \sqrt {c+d x}}{61440 d^4 e}+\frac {7 c \left (55 b^2 c^4+320 a b c^2 d^2+768 a^2 d^4\right ) (e x)^{7/2} \sqrt {c+d x}}{10240 d^3 e^2}+\frac {\left (55 b^2 c^4+320 a b c^2 d^2+768 a^2 d^4\right ) (e x)^{9/2} \sqrt {c+d x}}{3840 d^2 e^3}-\frac {b c \left (11 b c^2+64 a d^2\right ) (e x)^{5/2} (c+d x)^{7/2}}{384 d^4 e}+\frac {b \left (11 b c^2+64 a d^2\right ) (e x)^{7/2} (c+d x)^{7/2}}{224 d^3 e^2}-\frac {11 b^2 c (e x)^{9/2} (c+d x)^{7/2}}{144 d^2 e^3}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}+\frac {c^5 \left (55 b^2 c^4+320 a b c^2 d^2+768 a^2 d^4\right ) e^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{32768 d^{13/2}} \] Output:

-1/32768*c^4*(768*a^2*d^4+320*a*b*c^2*d^2+55*b^2*c^4)*e*(e*x)^(1/2)*(d*x+c 
)^(1/2)/d^6+1/49152*c^3*(768*a^2*d^4+320*a*b*c^2*d^2+55*b^2*c^4)*(e*x)^(3/ 
2)*(d*x+c)^(1/2)/d^5+31/61440*c^2*(768*a^2*d^4+320*a*b*c^2*d^2+55*b^2*c^4) 
*(e*x)^(5/2)*(d*x+c)^(1/2)/d^4/e+7/10240*c*(768*a^2*d^4+320*a*b*c^2*d^2+55 
*b^2*c^4)*(e*x)^(7/2)*(d*x+c)^(1/2)/d^3/e^2+1/3840*(768*a^2*d^4+320*a*b*c^ 
2*d^2+55*b^2*c^4)*(e*x)^(9/2)*(d*x+c)^(1/2)/d^2/e^3-1/384*b*c*(64*a*d^2+11 
*b*c^2)*(e*x)^(5/2)*(d*x+c)^(7/2)/d^4/e+1/224*b*(64*a*d^2+11*b*c^2)*(e*x)^ 
(7/2)*(d*x+c)^(7/2)/d^3/e^2-11/144*b^2*c*(e*x)^(9/2)*(d*x+c)^(7/2)/d^2/e^3 
+1/9*b^2*(e*x)^(11/2)*(d*x+c)^(7/2)/d/e^4+1/32768*c^5*(768*a^2*d^4+320*a*b 
*c^2*d^2+55*b^2*c^4)*e^(3/2)*arctanh(d^(1/2)*(e*x)^(1/2)/e^(1/2)/(d*x+c)^( 
1/2))/d^(13/2)
 

Mathematica [A] (verified)

Time = 2.04 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.65 \[ \int (e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=\frac {(e x)^{3/2} \left (\sqrt {d} \sqrt {x} \sqrt {c+d x} \left (16128 a^2 d^4 \left (-15 c^4+10 c^3 d x+248 c^2 d^2 x^2+336 c d^3 x^3+128 d^4 x^4\right )+960 a b d^2 \left (-105 c^6+70 c^5 d x-56 c^4 d^2 x^2+48 c^3 d^3 x^3+4736 c^2 d^4 x^4+7424 c d^5 x^5+3072 d^6 x^6\right )-5 b^2 \left (3465 c^8-2310 c^7 d x+1848 c^6 d^2 x^2-1584 c^5 d^3 x^3+1408 c^4 d^4 x^4-1280 c^3 d^5 x^5-316416 c^2 d^6 x^6-530432 c d^7 x^7-229376 d^8 x^8\right )\right )+630 c^5 \left (55 b^2 c^4+320 a b c^2 d^2+768 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{-\sqrt {c}+\sqrt {c+d x}}\right )\right )}{10321920 d^{13/2} x^{3/2}} \] Input:

Integrate[(e*x)^(3/2)*(c + d*x)^(5/2)*(a + b*x^2)^2,x]
 

Output:

((e*x)^(3/2)*(Sqrt[d]*Sqrt[x]*Sqrt[c + d*x]*(16128*a^2*d^4*(-15*c^4 + 10*c 
^3*d*x + 248*c^2*d^2*x^2 + 336*c*d^3*x^3 + 128*d^4*x^4) + 960*a*b*d^2*(-10 
5*c^6 + 70*c^5*d*x - 56*c^4*d^2*x^2 + 48*c^3*d^3*x^3 + 4736*c^2*d^4*x^4 + 
7424*c*d^5*x^5 + 3072*d^6*x^6) - 5*b^2*(3465*c^8 - 2310*c^7*d*x + 1848*c^6 
*d^2*x^2 - 1584*c^5*d^3*x^3 + 1408*c^4*d^4*x^4 - 1280*c^3*d^5*x^5 - 316416 
*c^2*d^6*x^6 - 530432*c*d^7*x^7 - 229376*d^8*x^8)) + 630*c^5*(55*b^2*c^4 + 
 320*a*b*c^2*d^2 + 768*a^2*d^4)*ArcTanh[(Sqrt[d]*Sqrt[x])/(-Sqrt[c] + Sqrt 
[c + d*x])]))/(10321920*d^(13/2)*x^(3/2))
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 385, normalized size of antiderivative = 0.80, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {521, 27, 2125, 27, 521, 27, 90, 60, 60, 60, 60, 60, 65, 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.

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

\(\Big \downarrow \) 521

\(\displaystyle \frac {\int \frac {1}{2} (e x)^{3/2} (c+d x)^{5/2} \left (-11 b^2 c x^3 e^4+36 a b d x^2 e^4+18 a^2 d e^4\right )dx}{9 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e x)^{3/2} (c+d x)^{5/2} \left (-11 b^2 c x^3 e^4+36 a b d x^2 e^4+18 a^2 d e^4\right )dx}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 2125

\(\displaystyle \frac {\frac {\int \frac {9}{2} e^7 (e x)^{3/2} (c+d x)^{5/2} \left (32 a^2 d^2+b \left (11 b c^2+64 a d^2\right ) x^2\right )dx}{8 d e^3}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {9 e^4 \int (e x)^{3/2} (c+d x)^{5/2} \left (32 a^2 d^2+b \left (11 b c^2+64 a d^2\right ) x^2\right )dx}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 521

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\int \frac {7}{2} e^2 (e x)^{3/2} (c+d x)^{5/2} \left (64 a^2 d^3-b c \left (11 b c^2+64 a d^2\right ) x\right )dx}{7 d e^2}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\int (e x)^{3/2} (c+d x)^{5/2} \left (64 a^2 d^3-b c \left (11 b c^2+64 a d^2\right ) x\right )dx}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\frac {\left (768 a^2 d^4+320 a b c^2 d^2+55 b^2 c^4\right ) \int (e x)^{3/2} (c+d x)^{5/2}dx}{12 d}-\frac {b c (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{6 d e}}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\frac {\left (768 a^2 d^4+320 a b c^2 d^2+55 b^2 c^4\right ) \left (\frac {1}{2} c \int (e x)^{3/2} (c+d x)^{3/2}dx+\frac {(e x)^{5/2} (c+d x)^{5/2}}{5 e}\right )}{12 d}-\frac {b c (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{6 d e}}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\frac {\left (768 a^2 d^4+320 a b c^2 d^2+55 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \int (e x)^{3/2} \sqrt {c+d x}dx+\frac {(e x)^{5/2} (c+d x)^{3/2}}{4 e}\right )+\frac {(e x)^{5/2} (c+d x)^{5/2}}{5 e}\right )}{12 d}-\frac {b c (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{6 d e}}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\frac {\left (768 a^2 d^4+320 a b c^2 d^2+55 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \int \frac {(e x)^{3/2}}{\sqrt {c+d x}}dx+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )+\frac {(e x)^{5/2} (c+d x)^{3/2}}{4 e}\right )+\frac {(e x)^{5/2} (c+d x)^{5/2}}{5 e}\right )}{12 d}-\frac {b c (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{6 d e}}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\frac {\left (768 a^2 d^4+320 a b c^2 d^2+55 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \int \frac {\sqrt {e x}}{\sqrt {c+d x}}dx}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )+\frac {(e x)^{5/2} (c+d x)^{3/2}}{4 e}\right )+\frac {(e x)^{5/2} (c+d x)^{5/2}}{5 e}\right )}{12 d}-\frac {b c (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{6 d e}}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\frac {\left (768 a^2 d^4+320 a b c^2 d^2+55 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{\sqrt {e x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )+\frac {(e x)^{5/2} (c+d x)^{3/2}}{4 e}\right )+\frac {(e x)^{5/2} (c+d x)^{5/2}}{5 e}\right )}{12 d}-\frac {b c (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{6 d e}}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\frac {\left (768 a^2 d^4+320 a b c^2 d^2+55 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c e \int \frac {1}{e-\frac {d e x}{c+d x}}d\frac {\sqrt {e x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )+\frac {(e x)^{5/2} (c+d x)^{3/2}}{4 e}\right )+\frac {(e x)^{5/2} (c+d x)^{5/2}}{5 e}\right )}{12 d}-\frac {b c (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{6 d e}}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {9 e^4 \left (\frac {\frac {\left (768 a^2 d^4+320 a b c^2 d^2+55 b^2 c^4\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \left (\frac {(e x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 c e \left (\frac {\sqrt {e x} \sqrt {c+d x}}{d}-\frac {c \sqrt {e} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e x}}{\sqrt {e} \sqrt {c+d x}}\right )}{d^{3/2}}\right )}{4 d}\right )+\frac {(e x)^{5/2} \sqrt {c+d x}}{3 e}\right )+\frac {(e x)^{5/2} (c+d x)^{3/2}}{4 e}\right )+\frac {(e x)^{5/2} (c+d x)^{5/2}}{5 e}\right )}{12 d}-\frac {b c (e x)^{5/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{6 d e}}{2 d}+\frac {b (e x)^{7/2} (c+d x)^{7/2} \left (64 a d^2+11 b c^2\right )}{7 d e^2}\right )}{16 d}-\frac {11 b^2 c e (e x)^{9/2} (c+d x)^{7/2}}{8 d}}{18 d e^4}+\frac {b^2 (e x)^{11/2} (c+d x)^{7/2}}{9 d e^4}\)

Input:

Int[(e*x)^(3/2)*(c + d*x)^(5/2)*(a + b*x^2)^2,x]
 

Output:

(b^2*(e*x)^(11/2)*(c + d*x)^(7/2))/(9*d*e^4) + ((-11*b^2*c*e*(e*x)^(9/2)*( 
c + d*x)^(7/2))/(8*d) + (9*e^4*((b*(11*b*c^2 + 64*a*d^2)*(e*x)^(7/2)*(c + 
d*x)^(7/2))/(7*d*e^2) + (-1/6*(b*c*(11*b*c^2 + 64*a*d^2)*(e*x)^(5/2)*(c + 
d*x)^(7/2))/(d*e) + ((55*b^2*c^4 + 320*a*b*c^2*d^2 + 768*a^2*d^4)*(((e*x)^ 
(5/2)*(c + d*x)^(5/2))/(5*e) + (c*(((e*x)^(5/2)*(c + d*x)^(3/2))/(4*e) + ( 
3*c*(((e*x)^(5/2)*Sqrt[c + d*x])/(3*e) + (c*(((e*x)^(3/2)*Sqrt[c + d*x])/( 
2*d) - (3*c*e*((Sqrt[e*x]*Sqrt[c + d*x])/d - (c*Sqrt[e]*ArcTanh[(Sqrt[d]*S 
qrt[e*x])/(Sqrt[e]*Sqrt[c + d*x])])/d^(3/2)))/(4*d)))/6))/8))/2))/(12*d))/ 
(2*d)))/(16*d))/(18*d*e^4)
 

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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 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 521

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

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 [A] (verified)

Time = 0.27 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {\left (-1146880 d^{8} b^{2} x^{8}-2652160 b^{2} c \,x^{7} d^{7}-2949120 a b \,d^{8} x^{6}-1582080 b^{2} c^{2} d^{6} x^{6}-7127040 a b c \,d^{7} x^{5}-6400 b^{2} c^{3} d^{5} x^{5}-2064384 a^{2} d^{8} x^{4}-4546560 a b \,c^{2} d^{6} x^{4}+7040 b^{2} c^{4} d^{4} x^{4}-5419008 a^{2} c \,d^{7} x^{3}-46080 a b \,c^{3} d^{5} x^{3}-7920 b^{2} c^{5} d^{3} x^{3}-3999744 a^{2} c^{2} d^{6} x^{2}+53760 a b \,c^{4} d^{4} x^{2}+9240 b^{2} c^{6} d^{2} x^{2}-161280 a^{2} c^{3} d^{5} x -67200 a b \,c^{5} d^{3} x -11550 b^{2} c^{7} d x +241920 a^{2} c^{4} d^{4}+100800 a b \,c^{6} d^{2}+17325 b^{2} c^{8}\right ) x \sqrt {d x +c}\, e^{2}}{10321920 d^{6} \sqrt {e x}}+\frac {c^{5} \left (768 a^{2} d^{4}+320 b \,c^{2} d^{2} a +55 b^{2} c^{4}\right ) \ln \left (\frac {\frac {1}{2} c e +d e x}{\sqrt {d e}}+\sqrt {d e \,x^{2}+c e x}\right ) e^{2} \sqrt {\left (d x +c \right ) e x}}{65536 d^{6} \sqrt {d e}\, \sqrt {e x}\, \sqrt {d x +c}}\)	\(371\)
default	\(\frac {\sqrt {e x}\, \sqrt {d x +c}\, e \left (2293760 b^{2} d^{8} x^{8} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+5304320 b^{2} c \,d^{7} x^{7} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+5898240 a b \,d^{8} x^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+3164160 b^{2} c^{2} d^{6} x^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}[phone] a b c \,d^{7} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+12800 b^{2} c^{3} d^{5} x^{5} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+4128768 a^{2} d^{8} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+9093120 a b \,c^{2} d^{6} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-14080 b^{2} c^{4} d^{4} x^{4} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}[phone] a^{2} c \,d^{7} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+92160 a b \,c^{3} d^{5} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+15840 b^{2} c^{5} d^{3} x^{3} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+7999488 a^{2} c^{2} d^{6} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-107520 a b \,c^{4} d^{4} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}-18480 b^{2} c^{6} d^{2} x^{2} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+241920 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a^{2} c^{5} d^{4} e +100800 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) a b \,c^{7} d^{2} e +17325 \ln \left (\frac {2 d e x +2 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}+c e}{2 \sqrt {d e}}\right ) b^{2} c^{9} e +322560 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a^{2} c^{3} d^{5} x +134400 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a b \,c^{5} d^{3} x +23100 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c^{7} d x -483840 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a^{2} c^{4} d^{4}-201600 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, a b \,c^{6} d^{2}-34650 \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}\, b^{2} c^{8}\right )}{20643840 d^{6} \sqrt {\left (d x +c \right ) e x}\, \sqrt {d e}}\)	\(736\)

Input:

int((e*x)^(3/2)*(d*x+c)^(5/2)*(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

-1/10321920/d^6*(-1146880*b^2*d^8*x^8-2652160*b^2*c*d^7*x^7-2949120*a*b*d^ 
8*x^6-1582080*b^2*c^2*d^6*x^6-7127040*a*b*c*d^7*x^5-6400*b^2*c^3*d^5*x^5-2 
064384*a^2*d^8*x^4-4546560*a*b*c^2*d^6*x^4+7040*b^2*c^4*d^4*x^4-5419008*a^ 
2*c*d^7*x^3-46080*a*b*c^3*d^5*x^3-7920*b^2*c^5*d^3*x^3-3999744*a^2*c^2*d^6 
*x^2+53760*a*b*c^4*d^4*x^2+9240*b^2*c^6*d^2*x^2-161280*a^2*c^3*d^5*x-67200 
*a*b*c^5*d^3*x-11550*b^2*c^7*d*x+241920*a^2*c^4*d^4+100800*a*b*c^6*d^2+173 
25*b^2*c^8)*x*(d*x+c)^(1/2)*e^2/(e*x)^(1/2)+1/65536*c^5/d^6*(768*a^2*d^4+3 
20*a*b*c^2*d^2+55*b^2*c^4)*ln((1/2*c*e+d*e*x)/(d*e)^(1/2)+(d*e*x^2+c*e*x)^ 
(1/2))/(d*e)^(1/2)*e^2*((d*x+c)*e*x)^(1/2)/(e*x)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.45 \[ \int (e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((e*x)^(3/2)*(d*x+c)^(5/2)*(b*x^2+a)^2,x, algorithm="fricas")
 

Output:

[1/20643840*(315*(55*b^2*c^9 + 320*a*b*c^7*d^2 + 768*a^2*c^5*d^4)*e*sqrt(e 
/d)*log(2*d*e*x + 2*sqrt(d*x + c)*sqrt(e*x)*d*sqrt(e/d) + c*e) + 2*(114688 
0*b^2*d^8*e*x^8 + 2652160*b^2*c*d^7*e*x^7 + 15360*(103*b^2*c^2*d^6 + 192*a 
*b*d^8)*e*x^6 + 1280*(5*b^2*c^3*d^5 + 5568*a*b*c*d^7)*e*x^5 - 128*(55*b^2* 
c^4*d^4 - 35520*a*b*c^2*d^6 - 16128*a^2*d^8)*e*x^4 + 144*(55*b^2*c^5*d^3 + 
 320*a*b*c^3*d^5 + 37632*a^2*c*d^7)*e*x^3 - 168*(55*b^2*c^6*d^2 + 320*a*b* 
c^4*d^4 - 23808*a^2*c^2*d^6)*e*x^2 + 210*(55*b^2*c^7*d + 320*a*b*c^5*d^3 + 
 768*a^2*c^3*d^5)*e*x - 315*(55*b^2*c^8 + 320*a*b*c^6*d^2 + 768*a^2*c^4*d^ 
4)*e)*sqrt(d*x + c)*sqrt(e*x))/d^6, -1/10321920*(315*(55*b^2*c^9 + 320*a*b 
*c^7*d^2 + 768*a^2*c^5*d^4)*e*sqrt(-e/d)*arctan(sqrt(d*x + c)*sqrt(e*x)*d* 
sqrt(-e/d)/(d*e*x + c*e)) - (1146880*b^2*d^8*e*x^8 + 2652160*b^2*c*d^7*e*x 
^7 + 15360*(103*b^2*c^2*d^6 + 192*a*b*d^8)*e*x^6 + 1280*(5*b^2*c^3*d^5 + 5 
568*a*b*c*d^7)*e*x^5 - 128*(55*b^2*c^4*d^4 - 35520*a*b*c^2*d^6 - 16128*a^2 
*d^8)*e*x^4 + 144*(55*b^2*c^5*d^3 + 320*a*b*c^3*d^5 + 37632*a^2*c*d^7)*e*x 
^3 - 168*(55*b^2*c^6*d^2 + 320*a*b*c^4*d^4 - 23808*a^2*c^2*d^6)*e*x^2 + 21 
0*(55*b^2*c^7*d + 320*a*b*c^5*d^3 + 768*a^2*c^3*d^5)*e*x - 315*(55*b^2*c^8 
 + 320*a*b*c^6*d^2 + 768*a^2*c^4*d^4)*e)*sqrt(d*x + c)*sqrt(e*x))/d^6]
 

Sympy [F(-1)]

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

integrate((e*x)**(3/2)*(d*x+c)**(5/2)*(b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((e*x)^(3/2)*(d*x+c)^(5/2)*(b*x^2+a)^2,x, algorithm="maxima")
 

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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1732 vs. \(2 (415) = 830\).

Time = 0.58 (sec) , antiderivative size = 1732, normalized size of antiderivative = 3.59 \[ \int (e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

-1/10321920*(161280*(105*c^4*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d* 
x + c)*d*e - c*d*e)))/(sqrt(d*e)*d^2) - sqrt((d*x + c)*d*e - c*d*e)*(2*(d* 
x + c)*(4*(d*x + c)*(6*(d*x + c)/d^3 - 25*c/d^3) + 163*c^2/d^3) - 279*c^3/ 
d^3)*sqrt(d*x + c))*a^2*c*abs(d) + 8064*(3465*c^6*e*log(abs(-sqrt(d*e)*sqr 
t(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e)*d^4) - sqrt((d*x + c 
)*d*e - c*d*e)*(2*(4*(2*(d*x + c)*(8*(d*x + c)*(10*(d*x + c)/d^5 - 61*c/d^ 
5) + 1251*c^2/d^5) - 3481*c^3/d^5)*(d*x + c) + 11395*c^4/d^5)*(d*x + c) - 
11895*c^5/d^5)*sqrt(d*x + c))*a*b*c*abs(d) + 270*(45045*c^8*e*log(abs(-sqr 
t(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e)*d^6) - sqr 
t((d*x + c)*d*e - c*d*e)*(2*(4*(2*(8*(2*(d*x + c)*(4*(d*x + c)*(14*(d*x + 
c)/d^7 - 113*c/d^7) + 1601*c^2/d^7) - 6513*c^3/d^7)*(d*x + c) + 66823*c^4/ 
d^7)*(d*x + c) - 111685*c^5/d^7)*(d*x + c) + 244391*c^6/d^7)*(d*x + c) - 1 
84331*c^7/d^7)*sqrt(d*x + c))*b^2*c*abs(d) + 107520*(105*c^4*e*log(abs(-sq 
rt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - c*d*e)))/(sqrt(d*e)*d^2) - sq 
rt((d*x + c)*d*e - c*d*e)*(2*(d*x + c)*(4*(d*x + c)*(6*(d*x + c)/d^3 - 25* 
c/d^3) + 163*c^2/d^3) - 279*c^3/d^3)*sqrt(d*x + c))*a*b*c^3*abs(d)/d^2 + 1 
344*(3465*c^6*e*log(abs(-sqrt(d*e)*sqrt(d*x + c) + sqrt((d*x + c)*d*e - c* 
d*e)))/(sqrt(d*e)*d^4) - sqrt((d*x + c)*d*e - c*d*e)*(2*(4*(2*(d*x + c)*(8 
*(d*x + c)*(10*(d*x + c)/d^5 - 61*c/d^5) + 1251*c^2/d^5) - 3481*c^3/d^5)*( 
d*x + c) + 11395*c^4/d^5)*(d*x + c) - 11895*c^5/d^5)*sqrt(d*x + c))*b^2...
 

Mupad [F(-1)]

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

int((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x)^(5/2),x)
 

Output:

int((e*x)^(3/2)*(a + b*x^2)^2*(c + d*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.78 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.08 \[ \int (e x)^{3/2} (c+d x)^{5/2} \left (a+b x^2\right )^2 \, dx=\frac {\sqrt {e}\, e \left (-241920 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d^{5}+161280 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{6} x +3999744 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{7} x^{2}+5419008 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{8} x^{3}+2064384 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{9} x^{4}-100800 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{6} d^{3}+67200 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} d^{4} x -53760 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{5} x^{2}+46080 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{6} x^{3}+4546560 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{7} x^{4}+7127040 \sqrt {x}\, \sqrt {d x +c}\, a b c \,d^{8} x^{5}+2949120 \sqrt {x}\, \sqrt {d x +c}\, a b \,d^{9} x^{6}-17325 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{8} d +11550 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{7} d^{2} x -9240 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} d^{3} x^{2}+7920 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d^{4} x^{3}-7040 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{5} x^{4}+6400 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{3} d^{6} x^{5}+1582080 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{2} d^{7} x^{6}+2652160 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c \,d^{8} x^{7}+1146880 \sqrt {x}\, \sqrt {d x +c}\, b^{2} d^{9} x^{8}+241920 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a^{2} c^{5} d^{4}+100800 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) a b \,c^{7} d^{2}+17325 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d x +c}+\sqrt {x}\, \sqrt {d}}{\sqrt {c}}\right ) b^{2} c^{9}\right )}{10321920 d^{7}} \] Input:

int((e*x)^(3/2)*(d*x+c)^(5/2)*(b*x^2+a)^2,x)
 

Output:

(sqrt(e)*e*( - 241920*sqrt(x)*sqrt(c + d*x)*a**2*c**4*d**5 + 161280*sqrt(x 
)*sqrt(c + d*x)*a**2*c**3*d**6*x + 3999744*sqrt(x)*sqrt(c + d*x)*a**2*c**2 
*d**7*x**2 + 5419008*sqrt(x)*sqrt(c + d*x)*a**2*c*d**8*x**3 + 2064384*sqrt 
(x)*sqrt(c + d*x)*a**2*d**9*x**4 - 100800*sqrt(x)*sqrt(c + d*x)*a*b*c**6*d 
**3 + 67200*sqrt(x)*sqrt(c + d*x)*a*b*c**5*d**4*x - 53760*sqrt(x)*sqrt(c + 
 d*x)*a*b*c**4*d**5*x**2 + 46080*sqrt(x)*sqrt(c + d*x)*a*b*c**3*d**6*x**3 
+ 4546560*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**7*x**4 + 7127040*sqrt(x)*sqrt( 
c + d*x)*a*b*c*d**8*x**5 + 2949120*sqrt(x)*sqrt(c + d*x)*a*b*d**9*x**6 - 1 
7325*sqrt(x)*sqrt(c + d*x)*b**2*c**8*d + 11550*sqrt(x)*sqrt(c + d*x)*b**2* 
c**7*d**2*x - 9240*sqrt(x)*sqrt(c + d*x)*b**2*c**6*d**3*x**2 + 7920*sqrt(x 
)*sqrt(c + d*x)*b**2*c**5*d**4*x**3 - 7040*sqrt(x)*sqrt(c + d*x)*b**2*c**4 
*d**5*x**4 + 6400*sqrt(x)*sqrt(c + d*x)*b**2*c**3*d**6*x**5 + 1582080*sqrt 
(x)*sqrt(c + d*x)*b**2*c**2*d**7*x**6 + 2652160*sqrt(x)*sqrt(c + d*x)*b**2 
*c*d**8*x**7 + 1146880*sqrt(x)*sqrt(c + d*x)*b**2*d**9*x**8 + 241920*sqrt( 
d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a**2*c**5*d**4 + 100800* 
sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*a*b*c**7*d**2 + 173 
25*sqrt(d)*log((sqrt(c + d*x) + sqrt(x)*sqrt(d))/sqrt(c))*b**2*c**9))/(103 
21920*d**7)