3.1201 \(\int \frac{(A+B x) (d+e x)}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=85 \[ \frac{2 B e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.127252, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 B e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.8314, size = 83, normalized size = 0.98 \[ \frac{2 B e \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} - \frac{2 \left (A b c d + x \left (2 A c^{2} d + B b^{2} e - b c \left (A e + B d\right )\right )\right )}{b^{2} c \sqrt{b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*e*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/c**(3/2) - 2*(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))

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Mathematica [A]  time = 0.149608, size = 102, normalized size = 1.2 \[ \frac{2 \left (\sqrt{c} (A c (-b d+b e x-2 c d x)+b B x (c d-b e))+b^2 B e \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{b^2 c^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 113, normalized size = 1.3 \[ -2\,{\frac{Ad \left ( 2\,cx+b \right ) }{{b}^{2}\sqrt{c{x}^{2}+bx}}}+2\,{\frac{xAe}{b\sqrt{c{x}^{2}+bx}}}+2\,{\frac{Bxd}{b\sqrt{c{x}^{2}+bx}}}-2\,{\frac{Bex}{c\sqrt{c{x}^{2}+bx}}}+{Be\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*d*(2*c*x+b)/b^2/(c*x^2+b*x)^(1/2)+2/b/(c*x^2+b*x)^(1/2)*x*A*e+2/b/(c*x^2+b*
x)^(1/2)*x*B*d-2*B*e/c/(c*x^2+b*x)^(1/2)*x+B*e/c^(3/2)*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)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303426, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c x^{2} + b x} B b^{2} e \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (A b c d -{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} e\right )} x\right )} \sqrt{c}}{\sqrt{c x^{2} + b x} b^{2} c^{\frac{3}{2}}}, \frac{2 \,{\left (\sqrt{c x^{2} + b x} B b^{2} e \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (A b c d -{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} e\right )} x\right )} \sqrt{-c}\right )}}{\sqrt{c x^{2} + b x} b^{2} \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.318023, size = 128, normalized size = 1.51 \[ -\frac{B e{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} - \frac{2 \,{\left (\frac{A d}{b} - \frac{{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + A b c e\right )} x}{b^{2} c}\right )}}{\sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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