3.1200 \(\int \frac{(A+B x) (d+e x)^2}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=151 \[ \frac{e \sqrt{b x+c x^2} \left (-2 b c (A e+B d)+4 A c^2 d+3 b^2 B e\right )}{b^2 c^2}-\frac{2 (d+e x) \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{c^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.37586, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{e \sqrt{b x+c x^2} \left (-2 b c (A e+B d)+4 A c^2 d+3 b^2 B e\right )}{b^2 c^2}-\frac{2 (d+e x) \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (2 A c e-3 b B e+4 B c d)}{c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^2)/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 41.1191, size = 153, normalized size = 1.01 \[ \frac{e \left (2 A c e - 3 B b e + 4 B c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} - \frac{2 \left (d + e x\right ) \left (A b c d + x \left (2 A c^{2} d + B b^{2} e - b c \left (A e + B d\right )\right )\right )}{b^{2} c \sqrt{b x + c x^{2}}} + \frac{e \sqrt{b x + c x^{2}} \left (4 A c^{2} d + 3 B b^{2} e - 2 b c \left (A e + B d\right )\right )}{b^{2} c^{2}} \]

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)**(3/2),x)

[Out]

e*(2*A*c*e - 3*B*b*e + 4*B*c*d)*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/c**(5/2) - 2
*(d + e*x)*(A*b*c*d + x*(2*A*c**2*d + B*b**2*e - b*c*(A*e + B*d)))/(b**2*c*sqrt(
b*x + c*x**2)) + e*sqrt(b*x + c*x**2)*(4*A*c**2*d + 3*B*b**2*e - 2*b*c*(A*e + B*
d))/(b**2*c**2)

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Mathematica [A]  time = 0.283353, size = 151, normalized size = 1. \[ \frac{\sqrt{c} \left (b B x \left (3 b^2 e^2+b c e (e x-4 d)+2 c^2 d^2\right )-2 A c \left (b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )\right )+b^2 e \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) (2 A c e-3 b B e+4 B c d)}{b^2 c^{5/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^2)/(b*x + c*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.016, size = 252, normalized size = 1.7 \[ -2\,{\frac{A{d}^{2} \left ( 2\,cx+b \right ) }{{b}^{2}\sqrt{c{x}^{2}+bx}}}-2\,{\frac{xA{e}^{2}}{c\sqrt{c{x}^{2}+bx}}}-4\,{\frac{Bxde}{c\sqrt{c{x}^{2}+bx}}}+{A{e}^{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}}+2\,{\frac{Bde}{{c}^{3/2}}\ln \left ({\frac{b/2+cx}{\sqrt{c}}}+\sqrt{c{x}^{2}+bx} \right ) }+4\,{\frac{xAde}{b\sqrt{c{x}^{2}+bx}}}+2\,{\frac{Bx{d}^{2}}{b\sqrt{c{x}^{2}+bx}}}+{\frac{B{e}^{2}{x}^{2}}{c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+3\,{\frac{B{e}^{2}bx}{{c}^{2}\sqrt{c{x}^{2}+bx}}}-{\frac{3\,B{e}^{2}b}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^2/(c*x^2+b*x)^(3/2),x)

[Out]

-2*A*d^2*(2*c*x+b)/b^2/(c*x^2+b*x)^(1/2)-2/c/(c*x^2+b*x)^(1/2)*x*A*e^2-4/c/(c*x^
2+b*x)^(1/2)*x*B*d*e+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*A*e^2+2
/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))*B*d*e+4/b/(c*x^2+b*x)^(1/2)*x
*A*d*e+2/b/(c*x^2+b*x)^(1/2)*x*B*d^2+B*e^2*x^2/c/(c*x^2+b*x)^(1/2)+3*B*e^2*b/c^2
/(c*x^2+b*x)^(1/2)*x-3/2*B*e^2*b/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2
))

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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)^2/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.325874, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (4 \, B b^{2} c d e -{\left (3 \, B b^{3} - 2 \, A b^{2} c\right )} e^{2}\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \,{\left (B b^{2} c e^{2} x^{2} - 2 \, A b c^{2} d^{2} +{\left (2 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \,{\left (B b^{2} c - A b c^{2}\right )} d e +{\left (3 \, B b^{3} - 2 \, A b^{2} c\right )} e^{2}\right )} x\right )} \sqrt{c}}{2 \, \sqrt{c x^{2} + b x} b^{2} c^{\frac{5}{2}}}, \frac{{\left (4 \, B b^{2} c d e -{\left (3 \, B b^{3} - 2 \, A b^{2} c\right )} e^{2}\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (B b^{2} c e^{2} x^{2} - 2 \, A b c^{2} d^{2} +{\left (2 \,{\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \,{\left (B b^{2} c - A b c^{2}\right )} d e +{\left (3 \, B b^{3} - 2 \, A b^{2} c\right )} e^{2}\right )} x\right )} \sqrt{-c}}{\sqrt{c x^{2} + b x} b^{2} \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^2/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )^{2}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**2/(c*x**2+b*x)**(3/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**2/(x*(b + c*x))**(3/2), x)

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GIAC/XCAS [A]  time = 0.313805, size = 209, normalized size = 1.38 \[ -\frac{\frac{2 \, A d^{2}}{b} -{\left (\frac{B x e^{2}}{c} + \frac{2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - 4 \, B b^{2} c d e + 4 \, A b c^{2} d e + 3 \, B b^{3} e^{2} - 2 \, A b^{2} c e^{2}}{b^{2} c^{2}}\right )} x}{\sqrt{c x^{2} + b x}} - \frac{{\left (4 \, B c d e - 3 \, B b e^{2} + 2 \, A c e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^2/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

-(2*A*d^2/b - (B*x*e^2/c + (2*B*b*c^2*d^2 - 4*A*c^3*d^2 - 4*B*b^2*c*d*e + 4*A*b*
c^2*d*e + 3*B*b^3*e^2 - 2*A*b^2*c*e^2)/(b^2*c^2))*x)/sqrt(c*x^2 + b*x) - 1/2*(4*
B*c*d*e - 3*B*b*e^2 + 2*A*c*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(
c) - b))/c^(5/2)