3.1183 \(\int (A+B x) (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=423 \[ \frac{\left (b x+c x^2\right )^{7/2} \left (14 c e x (18 A c e-11 b B e+4 B c d)+18 A c e (32 c d-9 b e)+B \left (99 b^2 e^2-324 b c d e+64 c^2 d^2\right )\right )}{2016 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{12288 c^5}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{768 c^4}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^{13/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^6}+\frac{B \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

[Out]

(5*b^4*(64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b^2*c*e*(2*B*d + A*e) - 32*b*c^2*d*(B*d
+ 2*A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(32768*c^6) - (5*b^2*(64*A*c^3*d^2 - 1
1*b^3*B*e^2 + 18*b^2*c*e*(2*B*d + A*e) - 32*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*(
b*x + c*x^2)^(3/2))/(12288*c^5) + ((64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b^2*c*e*(2*
B*d + A*e) - 32*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(768*c^4
) + (B*(d + e*x)^2*(b*x + c*x^2)^(7/2))/(9*c) + ((18*A*c*e*(32*c*d - 9*b*e) + B*
(64*c^2*d^2 - 324*b*c*d*e + 99*b^2*e^2) + 14*c*e*(4*B*c*d - 11*b*B*e + 18*A*c*e)
*x)*(b*x + c*x^2)^(7/2))/(2016*c^3) - (5*b^6*(64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b
^2*c*e*(2*B*d + A*e) - 32*b*c^2*d*(B*d + 2*A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x +
c*x^2]])/(32768*c^(13/2))

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Rubi [A]  time = 0.914511, antiderivative size = 423, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\left (b x+c x^2\right )^{7/2} \left (14 c e x (18 A c e-11 b B e+4 B c d)+18 A c e (32 c d-9 b e)+B \left (99 b^2 e^2-324 b c d e+64 c^2 d^2\right )\right )}{2016 c^3}-\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{12288 c^5}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{768 c^4}-\frac{5 b^6 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^{13/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} \left (18 b^2 c e (A e+2 B d)-32 b c^2 d (2 A e+B d)+64 A c^3 d^2-11 b^3 B e^2\right )}{32768 c^6}+\frac{B \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^2*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b^2*c*e*(2*B*d + A*e) - 32*b*c^2*d*(B*d
+ 2*A*e))*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(32768*c^6) - (5*b^2*(64*A*c^3*d^2 - 1
1*b^3*B*e^2 + 18*b^2*c*e*(2*B*d + A*e) - 32*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*(
b*x + c*x^2)^(3/2))/(12288*c^5) + ((64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b^2*c*e*(2*
B*d + A*e) - 32*b*c^2*d*(B*d + 2*A*e))*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(768*c^4
) + (B*(d + e*x)^2*(b*x + c*x^2)^(7/2))/(9*c) + ((18*A*c*e*(32*c*d - 9*b*e) + B*
(64*c^2*d^2 - 324*b*c*d*e + 99*b^2*e^2) + 14*c*e*(4*B*c*d - 11*b*B*e + 18*A*c*e)
*x)*(b*x + c*x^2)^(7/2))/(2016*c^3) - (5*b^6*(64*A*c^3*d^2 - 11*b^3*B*e^2 + 18*b
^2*c*e*(2*B*d + A*e) - 32*b*c^2*d*(B*d + 2*A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x +
c*x^2]])/(32768*c^(13/2))

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Rubi in Sympy [A]  time = 94.3586, size = 508, normalized size = 1.2 \[ \frac{B \left (d + e x\right )^{2} \left (b x + c x^{2}\right )^{\frac{7}{2}}}{9 c} - \frac{5 b^{6} \left (18 A b^{2} c e^{2} - 64 A b c^{2} d e + 64 A c^{3} d^{2} - 11 B b^{3} e^{2} + 36 B b^{2} c d e - 32 B b c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{32768 c^{\frac{13}{2}}} + \frac{5 b^{4} \left (b + 2 c x\right ) \sqrt{b x + c x^{2}} \left (18 A b^{2} c e^{2} - 64 A b c^{2} d e + 64 A c^{3} d^{2} - 11 B b^{3} e^{2} + 36 B b^{2} c d e - 32 B b c^{2} d^{2}\right )}{32768 c^{6}} - \frac{5 b^{2} \left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (18 A b^{2} c e^{2} - 64 A b c^{2} d e + 64 A c^{3} d^{2} - 11 B b^{3} e^{2} + 36 B b^{2} c d e - 32 B b c^{2} d^{2}\right )}{12288 c^{5}} + \frac{\left (b x + c x^{2}\right )^{\frac{7}{2}} \left (- \frac{81 A b c e^{2}}{2} + 144 A c^{2} d e + \frac{99 B b^{2} e^{2}}{4} - 81 B b c d e + 16 B c^{2} d^{2} + \frac{7 c e x \left (18 A c e - 11 B b e + 4 B c d\right )}{2}\right )}{504 c^{3}} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}} \left (18 A b^{2} c e^{2} - 64 A b c^{2} d e + 64 A c^{3} d^{2} - 11 B b^{3} e^{2} + 36 B b^{2} c d e - 32 B b c^{2} d^{2}\right )}{768 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**2*(c*x**2+b*x)**(5/2),x)

[Out]

B*(d + e*x)**2*(b*x + c*x**2)**(7/2)/(9*c) - 5*b**6*(18*A*b**2*c*e**2 - 64*A*b*c
**2*d*e + 64*A*c**3*d**2 - 11*B*b**3*e**2 + 36*B*b**2*c*d*e - 32*B*b*c**2*d**2)*
atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(32768*c**(13/2)) + 5*b**4*(b + 2*c*x)*sqrt(
b*x + c*x**2)*(18*A*b**2*c*e**2 - 64*A*b*c**2*d*e + 64*A*c**3*d**2 - 11*B*b**3*e
**2 + 36*B*b**2*c*d*e - 32*B*b*c**2*d**2)/(32768*c**6) - 5*b**2*(b + 2*c*x)*(b*x
+ c*x**2)**(3/2)*(18*A*b**2*c*e**2 - 64*A*b*c**2*d*e + 64*A*c**3*d**2 - 11*B*b*
*3*e**2 + 36*B*b**2*c*d*e - 32*B*b*c**2*d**2)/(12288*c**5) + (b*x + c*x**2)**(7/
2)*(-81*A*b*c*e**2/2 + 144*A*c**2*d*e + 99*B*b**2*e**2/4 - 81*B*b*c*d*e + 16*B*c
**2*d**2 + 7*c*e*x*(18*A*c*e - 11*B*b*e + 4*B*c*d)/2)/(504*c**3) + (b + 2*c*x)*(
b*x + c*x**2)**(5/2)*(18*A*b**2*c*e**2 - 64*A*b*c**2*d*e + 64*A*c**3*d**2 - 11*B
*b**3*e**2 + 36*B*b**2*c*d*e - 32*B*b*c**2*d**2)/(768*c**4)

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Mathematica [A]  time = 1.55899, size = 532, normalized size = 1.26 \[ \frac{(x (b+c x))^{5/2} \left (\frac{5 b^6 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (-18 b^2 c e (A e+2 B d)+32 b c^2 d (2 A e+B d)-64 A c^3 d^2+11 b^3 B e^2\right )}{c^{13/2} (b+c x)^{5/2}}+\frac{\sqrt{x} \left (210 b^7 c e (27 A e+54 B d+11 B e x)-84 b^6 c^2 \left (15 A e (16 d+3 e x)+2 B \left (60 d^2+45 d e x+11 e^2 x^2\right )\right )+48 b^5 c^3 \left (7 A \left (60 d^2+40 d e x+9 e^2 x^2\right )+B x \left (140 d^2+126 d e x+33 e^2 x^2\right )\right )-32 b^4 c^4 x \left (A \left (420 d^2+336 d e x+81 e^2 x^2\right )+2 B x \left (84 d^2+81 d e x+22 e^2 x^2\right )\right )+256 b^3 c^5 x^2 \left (A \left (42 d^2+36 d e x+9 e^2 x^2\right )+B x \left (18 d^2+18 d e x+5 e^2 x^2\right )\right )+1536 b^2 c^6 x^3 \left (A \left (378 d^2+592 d e x+243 e^2 x^2\right )+2 B x \left (148 d^2+243 d e x+103 e^2 x^2\right )\right )+2048 b c^7 x^4 \left (3 A \left (140 d^2+232 d e x+99 e^2 x^2\right )+B x \left (348 d^2+594 d e x+259 e^2 x^2\right )\right )+4096 c^8 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )-3465 b^8 B e^2\right )}{63 c^6 (b+c x)^2}\right )}{32768 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^2*(b*x + c*x^2)^(5/2),x]

[Out]

((x*(b + c*x))^(5/2)*((Sqrt[x]*(-3465*b^8*B*e^2 + 210*b^7*c*e*(54*B*d + 27*A*e +
11*B*e*x) + 256*b^3*c^5*x^2*(B*x*(18*d^2 + 18*d*e*x + 5*e^2*x^2) + A*(42*d^2 +
36*d*e*x + 9*e^2*x^2)) - 84*b^6*c^2*(15*A*e*(16*d + 3*e*x) + 2*B*(60*d^2 + 45*d*
e*x + 11*e^2*x^2)) + 4096*c^8*x^5*(3*A*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 2*B*x*
(36*d^2 + 63*d*e*x + 28*e^2*x^2)) + 48*b^5*c^3*(7*A*(60*d^2 + 40*d*e*x + 9*e^2*x
^2) + B*x*(140*d^2 + 126*d*e*x + 33*e^2*x^2)) - 32*b^4*c^4*x*(2*B*x*(84*d^2 + 81
*d*e*x + 22*e^2*x^2) + A*(420*d^2 + 336*d*e*x + 81*e^2*x^2)) + 1536*b^2*c^6*x^3*
(2*B*x*(148*d^2 + 243*d*e*x + 103*e^2*x^2) + A*(378*d^2 + 592*d*e*x + 243*e^2*x^
2)) + 2048*b*c^7*x^4*(3*A*(140*d^2 + 232*d*e*x + 99*e^2*x^2) + B*x*(348*d^2 + 59
4*d*e*x + 259*e^2*x^2))))/(63*c^6*(b + c*x)^2) + (5*b^6*(-64*A*c^3*d^2 + 11*b^3*
B*e^2 - 18*b^2*c*e*(2*B*d + A*e) + 32*b*c^2*d*(B*d + 2*A*e))*Log[c*Sqrt[x] + Sqr
t[c]*Sqrt[b + c*x]])/(c^(13/2)*(b + c*x)^(5/2))))/(32768*x^(5/2))

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Maple [B]  time = 0.02, size = 1227, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^2*(c*x^2+b*x)^(5/2),x)

[Out]

1/7*(c*x^2+b*x)^(7/2)/c*B*d^2+1/6*A*d^2*(c*x^2+b*x)^(5/2)*x-15/512*b^4/c^3*(c*x^
2+b*x)^(3/2)*x*B*d*e+45/4096*b^6/c^4*(c*x^2+b*x)^(1/2)*x*B*d*e-5/256*b^5/c^3*(c*
x^2+b*x)^(1/2)*x*A*d*e+5/96*b^3/c^2*(c*x^2+b*x)^(3/2)*x*A*d*e-9/112*b/c^2*(c*x^2
+b*x)^(7/2)*A*e^2-1/12*b/c*(c*x^2+b*x)^(5/2)*x*B*d^2-1/12*b^2/c^2*(c*x^2+b*x)^(5
/2)*A*d*e+5/256*A*d^2*b^4/c^2*(c*x^2+b*x)^(1/2)*x+45/8192*b^7/c^5*(c*x^2+b*x)^(1
/2)*B*d*e-45/16384*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d*e+
5/192*b^3/c^2*(c*x^2+b*x)^(3/2)*x*B*d^2+5/192*b^4/c^3*(c*x^2+b*x)^(3/2)*A*d*e-5/
512*b^5/c^3*(c*x^2+b*x)^(1/2)*x*B*d^2-5/512*b^6/c^4*(c*x^2+b*x)^(1/2)*A*d*e+5/10
24*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*d*e-11/144*B*e^2*b/c^
2*x*(c*x^2+b*x)^(7/2)-11/384*B*e^2*b^3/c^3*(c*x^2+b*x)^(5/2)*x+55/6144*B*e^2*b^5
/c^4*(c*x^2+b*x)^(3/2)*x-55/16384*B*e^2*b^7/c^5*(c*x^2+b*x)^(1/2)*x-15/1024*b^4/
c^3*(c*x^2+b*x)^(3/2)*x*A*e^2-15/1024*b^5/c^4*(c*x^2+b*x)^(3/2)*B*d*e-5/96*A*d^2
*b^2/c*(c*x^2+b*x)^(3/2)*x+1/4*x*(c*x^2+b*x)^(7/2)/c*B*d*e-9/56*b/c^2*(c*x^2+b*x
)^(7/2)*B*d*e+3/64*b^2/c^2*(c*x^2+b*x)^(5/2)*x*A*e^2+3/64*b^3/c^3*(c*x^2+b*x)^(5
/2)*B*d*e+45/8192*b^6/c^4*(c*x^2+b*x)^(1/2)*x*A*e^2-11/768*B*e^2*b^4/c^4*(c*x^2+
b*x)^(5/2)+55/12288*B*e^2*b^6/c^5*(c*x^2+b*x)^(3/2)-55/32768*B*e^2*b^8/c^6*(c*x^
2+b*x)^(1/2)+5/512*A*d^2*b^5/c^3*(c*x^2+b*x)^(1/2)-5/1024*A*d^2*b^6/c^(7/2)*ln((
1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+1/9*B*e^2*x^2*(c*x^2+b*x)^(7/2)/c+11/224*B
*e^2*b^2/c^3*(c*x^2+b*x)^(7/2)+55/65536*B*e^2*b^9/c^(13/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x)^(1/2))+2/7*(c*x^2+b*x)^(7/2)/c*A*d*e-1/24*b^2/c^2*(c*x^2+b*x)^(5/2
)*B*d^2+5/384*b^4/c^3*(c*x^2+b*x)^(3/2)*B*d^2-5/1024*b^6/c^4*(c*x^2+b*x)^(1/2)*B
*d^2+5/2048*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d^2+3/128*b^
3/c^3*(c*x^2+b*x)^(5/2)*A*e^2-15/2048*b^5/c^4*(c*x^2+b*x)^(3/2)*A*e^2+45/16384*b
^7/c^5*(c*x^2+b*x)^(1/2)*A*e^2-45/32768*b^8/c^(11/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x)^(1/2))*A*e^2+1/12*A*d^2/c*(c*x^2+b*x)^(5/2)*b-5/192*A*d^2*b^3/c^2*(c*x^2
+b*x)^(3/2)+1/8*x*(c*x^2+b*x)^(7/2)/c*A*e^2-1/6*b/c*(c*x^2+b*x)^(5/2)*x*A*d*e+3/
32*b^2/c^2*(c*x^2+b*x)^(5/2)*x*B*d*e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310256, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*(e*x + d)^2,x, algorithm="fricas")

[Out]

[1/4128768*(2*(229376*B*c^8*e^2*x^8 + 14336*(36*B*c^8*d*e + (37*B*b*c^7 + 18*A*c
^8)*e^2)*x^7 + 3072*(96*B*c^8*d^2 + 12*(33*B*b*c^7 + 16*A*c^8)*d*e + (103*B*b^2*
c^6 + 198*A*b*c^7)*e^2)*x^6 + 256*(96*(29*B*b*c^7 + 14*A*c^8)*d^2 + 12*(243*B*b^
2*c^6 + 464*A*b*c^7)*d*e + (5*B*b^3*c^5 + 1458*A*b^2*c^6)*e^2)*x^5 + 128*(96*(37
*B*b^2*c^6 + 70*A*b*c^7)*d^2 + 12*(3*B*b^3*c^5 + 592*A*b^2*c^6)*d*e - (11*B*b^4*
c^4 - 18*A*b^3*c^5)*e^2)*x^4 + 144*(32*(B*b^3*c^5 + 126*A*b^2*c^6)*d^2 - 4*(9*B*
b^4*c^4 - 16*A*b^3*c^5)*d*e + (11*B*b^5*c^3 - 18*A*b^4*c^4)*e^2)*x^3 - 10080*(B*
b^6*c^2 - 2*A*b^5*c^3)*d^2 + 1260*(9*B*b^7*c - 16*A*b^6*c^2)*d*e - 315*(11*B*b^8
- 18*A*b^7*c)*e^2 - 168*(32*(B*b^4*c^4 - 2*A*b^3*c^5)*d^2 - 4*(9*B*b^5*c^3 - 16
*A*b^4*c^4)*d*e + (11*B*b^6*c^2 - 18*A*b^5*c^3)*e^2)*x^2 + 210*(32*(B*b^5*c^3 -
2*A*b^4*c^4)*d^2 - 4*(9*B*b^6*c^2 - 16*A*b^5*c^3)*d*e + (11*B*b^7*c - 18*A*b^6*c
^2)*e^2)*x)*sqrt(c*x^2 + b*x)*sqrt(c) + 315*(32*(B*b^7*c^2 - 2*A*b^6*c^3)*d^2 -
4*(9*B*b^8*c - 16*A*b^7*c^2)*d*e + (11*B*b^9 - 18*A*b^8*c)*e^2)*log((2*c*x + b)*
sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(13/2), 1/2064384*((229376*B*c^8*e^2*x^8 + 1
4336*(36*B*c^8*d*e + (37*B*b*c^7 + 18*A*c^8)*e^2)*x^7 + 3072*(96*B*c^8*d^2 + 12*
(33*B*b*c^7 + 16*A*c^8)*d*e + (103*B*b^2*c^6 + 198*A*b*c^7)*e^2)*x^6 + 256*(96*(
29*B*b*c^7 + 14*A*c^8)*d^2 + 12*(243*B*b^2*c^6 + 464*A*b*c^7)*d*e + (5*B*b^3*c^5
+ 1458*A*b^2*c^6)*e^2)*x^5 + 128*(96*(37*B*b^2*c^6 + 70*A*b*c^7)*d^2 + 12*(3*B*
b^3*c^5 + 592*A*b^2*c^6)*d*e - (11*B*b^4*c^4 - 18*A*b^3*c^5)*e^2)*x^4 + 144*(32*
(B*b^3*c^5 + 126*A*b^2*c^6)*d^2 - 4*(9*B*b^4*c^4 - 16*A*b^3*c^5)*d*e + (11*B*b^5
*c^3 - 18*A*b^4*c^4)*e^2)*x^3 - 10080*(B*b^6*c^2 - 2*A*b^5*c^3)*d^2 + 1260*(9*B*
b^7*c - 16*A*b^6*c^2)*d*e - 315*(11*B*b^8 - 18*A*b^7*c)*e^2 - 168*(32*(B*b^4*c^4
- 2*A*b^3*c^5)*d^2 - 4*(9*B*b^5*c^3 - 16*A*b^4*c^4)*d*e + (11*B*b^6*c^2 - 18*A*
b^5*c^3)*e^2)*x^2 + 210*(32*(B*b^5*c^3 - 2*A*b^4*c^4)*d^2 - 4*(9*B*b^6*c^2 - 16*
A*b^5*c^3)*d*e + (11*B*b^7*c - 18*A*b^6*c^2)*e^2)*x)*sqrt(c*x^2 + b*x)*sqrt(-c)
+ 315*(32*(B*b^7*c^2 - 2*A*b^6*c^3)*d^2 - 4*(9*B*b^8*c - 16*A*b^7*c^2)*d*e + (11
*B*b^9 - 18*A*b^8*c)*e^2)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^
6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right ) \left (d + e x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**2*(c*x**2+b*x)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(A + B*x)*(d + e*x)**2, x)

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GIAC/XCAS [A]  time = 0.289878, size = 919, normalized size = 2.17 \[ \frac{1}{2064384} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \, B c^{2} x e^{2} + \frac{36 \, B c^{10} d e + 37 \, B b c^{9} e^{2} + 18 \, A c^{10} e^{2}}{c^{8}}\right )} x + \frac{3 \,{\left (96 \, B c^{10} d^{2} + 396 \, B b c^{9} d e + 192 \, A c^{10} d e + 103 \, B b^{2} c^{8} e^{2} + 198 \, A b c^{9} e^{2}\right )}}{c^{8}}\right )} x + \frac{2784 \, B b c^{9} d^{2} + 1344 \, A c^{10} d^{2} + 2916 \, B b^{2} c^{8} d e + 5568 \, A b c^{9} d e + 5 \, B b^{3} c^{7} e^{2} + 1458 \, A b^{2} c^{8} e^{2}}{c^{8}}\right )} x + \frac{3552 \, B b^{2} c^{8} d^{2} + 6720 \, A b c^{9} d^{2} + 36 \, B b^{3} c^{7} d e + 7104 \, A b^{2} c^{8} d e - 11 \, B b^{4} c^{6} e^{2} + 18 \, A b^{3} c^{7} e^{2}}{c^{8}}\right )} x + \frac{9 \,{\left (32 \, B b^{3} c^{7} d^{2} + 4032 \, A b^{2} c^{8} d^{2} - 36 \, B b^{4} c^{6} d e + 64 \, A b^{3} c^{7} d e + 11 \, B b^{5} c^{5} e^{2} - 18 \, A b^{4} c^{6} e^{2}\right )}}{c^{8}}\right )} x - \frac{21 \,{\left (32 \, B b^{4} c^{6} d^{2} - 64 \, A b^{3} c^{7} d^{2} - 36 \, B b^{5} c^{5} d e + 64 \, A b^{4} c^{6} d e + 11 \, B b^{6} c^{4} e^{2} - 18 \, A b^{5} c^{5} e^{2}\right )}}{c^{8}}\right )} x + \frac{105 \,{\left (32 \, B b^{5} c^{5} d^{2} - 64 \, A b^{4} c^{6} d^{2} - 36 \, B b^{6} c^{4} d e + 64 \, A b^{5} c^{5} d e + 11 \, B b^{7} c^{3} e^{2} - 18 \, A b^{6} c^{4} e^{2}\right )}}{c^{8}}\right )} x - \frac{315 \,{\left (32 \, B b^{6} c^{4} d^{2} - 64 \, A b^{5} c^{5} d^{2} - 36 \, B b^{7} c^{3} d e + 64 \, A b^{6} c^{4} d e + 11 \, B b^{8} c^{2} e^{2} - 18 \, A b^{7} c^{3} e^{2}\right )}}{c^{8}}\right )} - \frac{5 \,{\left (32 \, B b^{7} c^{2} d^{2} - 64 \, A b^{6} c^{3} d^{2} - 36 \, B b^{8} c d e + 64 \, A b^{7} c^{2} d e + 11 \, B b^{9} e^{2} - 18 \, A b^{8} c e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{65536 \, c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(B*x + A)*(e*x + d)^2,x, algorithm="giac")

[Out]

1/2064384*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(4*(14*(16*B*c^2*x*e^2 + (36*B*c^10*d
*e + 37*B*b*c^9*e^2 + 18*A*c^10*e^2)/c^8)*x + 3*(96*B*c^10*d^2 + 396*B*b*c^9*d*e
+ 192*A*c^10*d*e + 103*B*b^2*c^8*e^2 + 198*A*b*c^9*e^2)/c^8)*x + (2784*B*b*c^9*
d^2 + 1344*A*c^10*d^2 + 2916*B*b^2*c^8*d*e + 5568*A*b*c^9*d*e + 5*B*b^3*c^7*e^2
+ 1458*A*b^2*c^8*e^2)/c^8)*x + (3552*B*b^2*c^8*d^2 + 6720*A*b*c^9*d^2 + 36*B*b^3
*c^7*d*e + 7104*A*b^2*c^8*d*e - 11*B*b^4*c^6*e^2 + 18*A*b^3*c^7*e^2)/c^8)*x + 9*
(32*B*b^3*c^7*d^2 + 4032*A*b^2*c^8*d^2 - 36*B*b^4*c^6*d*e + 64*A*b^3*c^7*d*e + 1
1*B*b^5*c^5*e^2 - 18*A*b^4*c^6*e^2)/c^8)*x - 21*(32*B*b^4*c^6*d^2 - 64*A*b^3*c^7
*d^2 - 36*B*b^5*c^5*d*e + 64*A*b^4*c^6*d*e + 11*B*b^6*c^4*e^2 - 18*A*b^5*c^5*e^2
)/c^8)*x + 105*(32*B*b^5*c^5*d^2 - 64*A*b^4*c^6*d^2 - 36*B*b^6*c^4*d*e + 64*A*b^
5*c^5*d*e + 11*B*b^7*c^3*e^2 - 18*A*b^6*c^4*e^2)/c^8)*x - 315*(32*B*b^6*c^4*d^2
- 64*A*b^5*c^5*d^2 - 36*B*b^7*c^3*d*e + 64*A*b^6*c^4*d*e + 11*B*b^8*c^2*e^2 - 18
*A*b^7*c^3*e^2)/c^8) - 5/65536*(32*B*b^7*c^2*d^2 - 64*A*b^6*c^3*d^2 - 36*B*b^8*c
*d*e + 64*A*b^7*c^2*d*e + 11*B*b^9*e^2 - 18*A*b^8*c*e^2)*ln(abs(-2*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(13/2)