3.1206 \(\int \frac{(A+B x) (d+e x)^4}{\left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=341 \[ -\frac{2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac{e \sqrt{b x+c x^2} \left (2 b^3 c e^2 (3 A e+7 B d)+4 b^2 c^2 d e (A e+2 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{3 b^4 c^3}-\frac{2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt{b x+c x^2}}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}} \]

[Out]

(-2*(d + e*x)^3*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^2*c*
(b*x + c*x^2)^(3/2)) - (2*(d + e*x)*(b*c*d^2*(4*b*B*c*d - 8*A*c^2*d - b^2*B*e +
10*A*b*c*e) - (16*A*c^4*d^3 - 5*b^4*B*e^3 + 4*b^2*c^2*d*e*(B*d + A*e) + 2*b^3*c*
e^2*(3*B*d + A*e) - 8*b*c^3*d^2*(B*d + 3*A*e))*x))/(3*b^4*c^2*Sqrt[b*x + c*x^2])
- (e*(32*A*c^4*d^3 - 15*b^4*B*e^3 + 4*b^2*c^2*d*e*(2*B*d + A*e) - 16*b*c^3*d^2*
(B*d + 3*A*e) + 2*b^3*c*e^2*(7*B*d + 3*A*e))*Sqrt[b*x + c*x^2])/(3*b^4*c^3) + (e
^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(7/2)

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Rubi [A]  time = 1.06988, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 (d+e x)^3 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac{e \sqrt{b x+c x^2} \left (2 b^3 c e^2 (3 A e+7 B d)+4 b^2 c^2 d e (A e+2 B d)-16 b c^3 d^2 (3 A e+B d)+32 A c^4 d^3-15 b^4 B e^3\right )}{3 b^4 c^3}-\frac{2 (d+e x) \left (b c d^2 \left (10 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-x \left (2 b^3 c e^2 (A e+3 B d)+4 b^2 c^2 d e (A e+B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-5 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt{b x+c x^2}}+\frac{e^3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^3*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^2*c*
(b*x + c*x^2)^(3/2)) - (2*(d + e*x)*(b*c*d^2*(4*b*B*c*d - 8*A*c^2*d - b^2*B*e +
10*A*b*c*e) - (16*A*c^4*d^3 - 5*b^4*B*e^3 + 4*b^2*c^2*d*e*(B*d + A*e) + 2*b^3*c*
e^2*(3*B*d + A*e) - 8*b*c^3*d^2*(B*d + 3*A*e))*x))/(3*b^4*c^2*Sqrt[b*x + c*x^2])
- (e*(32*A*c^4*d^3 - 15*b^4*B*e^3 + 4*b^2*c^2*d*e*(2*B*d + A*e) - 16*b*c^3*d^2*
(B*d + 3*A*e) + 2*b^3*c*e^2*(7*B*d + 3*A*e))*Sqrt[b*x + c*x^2])/(3*b^4*c^3) + (e
^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/c^(7/2)

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Rubi in Sympy [A]  time = 156.265, size = 405, normalized size = 1.19 \[ \frac{e^{3} \left (2 A c e - 5 B b e + 8 B c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{7}{2}}} - \frac{2 \left (d + e x\right )^{3} \left (A b c d + x \left (2 A c^{2} d + B b^{2} e - b c \left (A e + B d\right )\right )\right )}{3 b^{2} c \left (b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (d + e x\right ) \left (\frac{b c d^{2} \left (- 10 A b c e + 8 A c^{2} d + B b^{2} e - 4 B b c d\right )}{2} + x \left (A b^{3} c e^{3} + 2 A b^{2} c^{2} d e^{2} - 12 A b c^{3} d^{2} e + 8 A c^{4} d^{3} - \frac{5 B b^{4} e^{3}}{2} + 3 B b^{3} c d e^{2} + 2 B b^{2} c^{2} d^{2} e - 4 B b c^{3} d^{3}\right )\right )}{3 b^{4} c^{2} \sqrt{b x + c x^{2}}} - \frac{e \sqrt{b x + c x^{2}} \left (6 A b^{3} c e^{3} + 4 A b^{2} c^{2} d e^{2} - 48 A b c^{3} d^{2} e + 32 A c^{4} d^{3} - 15 B b^{4} e^{3} + 14 B b^{3} c d e^{2} + 8 B b^{2} c^{2} d^{2} e - 16 B b c^{3} d^{3}\right )}{3 b^{4} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**4/(c*x**2+b*x)**(5/2),x)

[Out]

e**3*(2*A*c*e - 5*B*b*e + 8*B*c*d)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/c**(7/2)
- 2*(d + e*x)**3*(A*b*c*d + x*(2*A*c**2*d + B*b**2*e - b*c*(A*e + B*d)))/(3*b**2
*c*(b*x + c*x**2)**(3/2)) + 4*(d + e*x)*(b*c*d**2*(-10*A*b*c*e + 8*A*c**2*d + B*
b**2*e - 4*B*b*c*d)/2 + x*(A*b**3*c*e**3 + 2*A*b**2*c**2*d*e**2 - 12*A*b*c**3*d*
*2*e + 8*A*c**4*d**3 - 5*B*b**4*e**3/2 + 3*B*b**3*c*d*e**2 + 2*B*b**2*c**2*d**2*
e - 4*B*b*c**3*d**3))/(3*b**4*c**2*sqrt(b*x + c*x**2)) - e*sqrt(b*x + c*x**2)*(6
*A*b**3*c*e**3 + 4*A*b**2*c**2*d*e**2 - 48*A*b*c**3*d**2*e + 32*A*c**4*d**3 - 15
*B*b**4*e**3 + 14*B*b**3*c*d*e**2 + 8*B*b**2*c**2*d**2*e - 16*B*b*c**3*d**3)/(3*
b**4*c**3)

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Mathematica [A]  time = 0.640568, size = 237, normalized size = 0.7 \[ \frac{x^{5/2} \left (\frac{e^3 (b+c x)^{5/2} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) (2 A c e-5 b B e+8 B c d)}{c^{7/2}}-\frac{(b+c x) \left (2 x^2 (b+c x) (c d-b e)^3 \left (b c (5 B d-4 A e)-8 A c^2 d+7 b^2 B e\right )+2 c^3 d^3 x (b+c x)^2 (3 b (4 A e+B d)-8 A c d)+2 b x^2 (b B-A c) (c d-b e)^4+2 A b c^3 d^4 (b+c x)^2-3 b^4 B e^4 x^2 (b+c x)^2\right )}{3 b^4 c^3 x^{3/2}}\right )}{(x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^4)/(b*x + c*x^2)^(5/2),x]

[Out]

(x^(5/2)*(-((b + c*x)*(2*b*(b*B - A*c)*(c*d - b*e)^4*x^2 + 2*(c*d - b*e)^3*(-8*A
*c^2*d + 7*b^2*B*e + b*c*(5*B*d - 4*A*e))*x^2*(b + c*x) + 2*A*b*c^3*d^4*(b + c*x
)^2 + 2*c^3*d^3*(-8*A*c*d + 3*b*(B*d + 4*A*e))*x*(b + c*x)^2 - 3*b^4*B*e^4*x^2*(
b + c*x)^2))/(3*b^4*c^3*x^(3/2)) + (e^3*(8*B*c*d - 5*b*B*e + 2*A*c*e)*(b + c*x)^
(5/2)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/c^(7/2)))/(x*(b + c*x))^(5/2)

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Maple [B]  time = 0.017, size = 1026, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^4/(c*x^2+b*x)^(5/2),x)

[Out]

-4/3*x^3/c/(c*x^2+b*x)^(3/2)*B*d*e^3+5/6*B*e^4*b/c^2*x^3/(c*x^2+b*x)^(3/2)-5/4*B
*e^4*b^2/c^3*x^2/(c*x^2+b*x)^(3/2)-5/12*B*e^4*b^3/c^4/(c*x^2+b*x)^(3/2)*x+2/c^2/
(c*x^2+b*x)^(1/2)*B*d^2*e^2+2/3/b/(c*x^2+b*x)^(3/2)*x*B*d^4-32/3/b^2/(c*x^2+b*x)
^(1/2)*A*d^3*e+5/12*B*e^4*b^2/c^4/(c*x^2+b*x)^(1/2)+4/3/c^2/(c*x^2+b*x)^(1/2)*A*
d*e^3-5/2*B*e^4*b/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+16/3*A*d^4*c
/b^3/(c*x^2+b*x)^(1/2)+B*e^4*x^4/c/(c*x^2+b*x)^(3/2)-1/6*b/c^3/(c*x^2+b*x)^(1/2)
*A*e^4+4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d*e^3+4/b/c/(c*x^2+
b*x)^(1/2)*x*B*d^2*e^2-64/3/b^3*c/(c*x^2+b*x)^(1/2)*x*A*d^3*e-4/3*b/c^2/(c*x^2+b
*x)^(3/2)*x*A*d*e^3+1/2*b/c^2*x^2/(c*x^2+b*x)^(3/2)*A*e^4+1/6*b^2/c^3/(c*x^2+b*x
)^(3/2)*x*A*e^4-28/3/c^2/(c*x^2+b*x)^(1/2)*x*B*d*e^3-2/3*b/c^3/(c*x^2+b*x)^(1/2)
*B*d*e^3+8/3/b/c/(c*x^2+b*x)^(1/2)*B*d^3*e-8/3/b^2/(c*x^2+b*x)^(1/2)*B*d^4-2/3*A
*d^4/b/(c*x^2+b*x)^(3/2)+1/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*e
^4-1/3*x^3/c/(c*x^2+b*x)^(3/2)*A*e^4+35/6*B*e^4*b/c^3/(c*x^2+b*x)^(1/2)*x+8/3/b/
(c*x^2+b*x)^(3/2)*x*A*d^3*e-16/3/b^3*c/(c*x^2+b*x)^(1/2)*x*B*d^4-4/c/(c*x^2+b*x)
^(3/2)*x*A*d^2*e^2-8/3/c/(c*x^2+b*x)^(3/2)*x*B*d^3*e+8/b^2/(c*x^2+b*x)^(1/2)*x*A
*d^2*e^2+16/3/b^2/(c*x^2+b*x)^(1/2)*x*B*d^3*e+4/b/c/(c*x^2+b*x)^(1/2)*A*d^2*e^2-
4*x^2/c/(c*x^2+b*x)^(3/2)*A*d*e^3-4/3*A*d^4/b^2/(c*x^2+b*x)^(3/2)*c*x+32/3*A*d^4
*c^2/b^4/(c*x^2+b*x)^(1/2)*x-6*x^2/c/(c*x^2+b*x)^(3/2)*B*d^2*e^2-2*b/c^2/(c*x^2+
b*x)^(3/2)*x*B*d^2*e^2+8/3/b/c/(c*x^2+b*x)^(1/2)*x*A*d*e^3+2*b/c^2*x^2/(c*x^2+b*
x)^(3/2)*B*d*e^3+2/3*b^2/c^3/(c*x^2+b*x)^(3/2)*x*B*d*e^3-7/3/c^2/(c*x^2+b*x)^(1/
2)*x*A*e^4

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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((B*x + A)*(e*x + d)^4/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298283, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(3*((8*B*b^4*c^2*d*e^3 - (5*B*b^5*c - 2*A*b^4*c^2)*e^4)*x^2 + (8*B*b^5*c*d*
e^3 - (5*B*b^6 - 2*A*b^5*c)*e^4)*x)*sqrt(c*x^2 + b*x)*log((2*c*x + b)*sqrt(c) +
2*sqrt(c*x^2 + b*x)*c) + 2*(3*B*b^4*c^2*e^4*x^4 - 2*A*b^3*c^3*d^4 - 4*(4*(B*b*c^
5 - 2*A*c^6)*d^4 - 4*(B*b^2*c^4 - 4*A*b*c^5)*d^3*e - 3*(B*b^3*c^3 + 2*A*b^2*c^4)
*d^2*e^2 + 2*(4*B*b^4*c^2 - A*b^3*c^3)*d*e^3 - (5*B*b^5*c - 2*A*b^4*c^2)*e^4)*x^
3 + 3*(12*A*b^3*c^3*d^2*e^2 - 8*B*b^5*c*d*e^3 - 8*(B*b^2*c^4 - 2*A*b*c^5)*d^4 +
8*(B*b^3*c^3 - 4*A*b^2*c^4)*d^3*e + (5*B*b^6 - 2*A*b^5*c)*e^4)*x^2 - 6*(4*A*b^3*
c^3*d^3*e + (B*b^3*c^3 - 2*A*b^2*c^4)*d^4)*x)*sqrt(c))/((b^4*c^4*x^2 + b^5*c^3*x
)*sqrt(c*x^2 + b*x)*sqrt(c)), 1/3*(3*((8*B*b^4*c^2*d*e^3 - (5*B*b^5*c - 2*A*b^4*
c^2)*e^4)*x^2 + (8*B*b^5*c*d*e^3 - (5*B*b^6 - 2*A*b^5*c)*e^4)*x)*sqrt(c*x^2 + b*
x)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (3*B*b^4*c^2*e^4*x^4 - 2*A*b^3*c^3
*d^4 - 4*(4*(B*b*c^5 - 2*A*c^6)*d^4 - 4*(B*b^2*c^4 - 4*A*b*c^5)*d^3*e - 3*(B*b^3
*c^3 + 2*A*b^2*c^4)*d^2*e^2 + 2*(4*B*b^4*c^2 - A*b^3*c^3)*d*e^3 - (5*B*b^5*c - 2
*A*b^4*c^2)*e^4)*x^3 + 3*(12*A*b^3*c^3*d^2*e^2 - 8*B*b^5*c*d*e^3 - 8*(B*b^2*c^4
- 2*A*b*c^5)*d^4 + 8*(B*b^3*c^3 - 4*A*b^2*c^4)*d^3*e + (5*B*b^6 - 2*A*b^5*c)*e^4
)*x^2 - 6*(4*A*b^3*c^3*d^3*e + (B*b^3*c^3 - 2*A*b^2*c^4)*d^4)*x)*sqrt(-c))/((b^4
*c^4*x^2 + b^5*c^3*x)*sqrt(c*x^2 + b*x)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{4}}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**4/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**4/(x*(b + c*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.302718, size = 500, normalized size = 1.47 \[ -\frac{\frac{2 \, A d^{4}}{b} -{\left ({\left ({\left (\frac{3 \, B x e^{4}}{c} - \frac{4 \,{\left (4 \, B b c^{5} d^{4} - 8 \, A c^{6} d^{4} - 4 \, B b^{2} c^{4} d^{3} e + 16 \, A b c^{5} d^{3} e - 3 \, B b^{3} c^{3} d^{2} e^{2} - 6 \, A b^{2} c^{4} d^{2} e^{2} + 8 \, B b^{4} c^{2} d e^{3} - 2 \, A b^{3} c^{3} d e^{3} - 5 \, B b^{5} c e^{4} + 2 \, A b^{4} c^{2} e^{4}\right )}}{b^{4} c^{3}}\right )} x - \frac{3 \,{\left (8 \, B b^{2} c^{4} d^{4} - 16 \, A b c^{5} d^{4} - 8 \, B b^{3} c^{3} d^{3} e + 32 \, A b^{2} c^{4} d^{3} e - 12 \, A b^{3} c^{3} d^{2} e^{2} + 8 \, B b^{5} c d e^{3} - 5 \, B b^{6} e^{4} + 2 \, A b^{5} c e^{4}\right )}}{b^{4} c^{3}}\right )} x - \frac{6 \,{\left (B b^{3} c^{3} d^{4} - 2 \, A b^{2} c^{4} d^{4} + 4 \, A b^{3} c^{3} d^{3} e\right )}}{b^{4} c^{3}}\right )} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} - \frac{{\left (8 \, B c d e^{3} - 5 \, B b e^{4} + 2 \, A c e^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(c*x^2 + b*x)^(5/2),x, algorithm="giac")

[Out]

-1/3*(2*A*d^4/b - (((3*B*x*e^4/c - 4*(4*B*b*c^5*d^4 - 8*A*c^6*d^4 - 4*B*b^2*c^4*
d^3*e + 16*A*b*c^5*d^3*e - 3*B*b^3*c^3*d^2*e^2 - 6*A*b^2*c^4*d^2*e^2 + 8*B*b^4*c
^2*d*e^3 - 2*A*b^3*c^3*d*e^3 - 5*B*b^5*c*e^4 + 2*A*b^4*c^2*e^4)/(b^4*c^3))*x - 3
*(8*B*b^2*c^4*d^4 - 16*A*b*c^5*d^4 - 8*B*b^3*c^3*d^3*e + 32*A*b^2*c^4*d^3*e - 12
*A*b^3*c^3*d^2*e^2 + 8*B*b^5*c*d*e^3 - 5*B*b^6*e^4 + 2*A*b^5*c*e^4)/(b^4*c^3))*x
- 6*(B*b^3*c^3*d^4 - 2*A*b^2*c^4*d^4 + 4*A*b^3*c^3*d^3*e)/(b^4*c^3))*x)/(c*x^2
+ b*x)^(3/2) - 1/2*(8*B*c*d*e^3 - 5*B*b*e^4 + 2*A*c*e^4)*ln(abs(-2*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(7/2)