3.366 \(\int \frac{1}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx\)

Optimal. Leaf size=102 \[ \frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}-\frac{2 e}{d \sqrt{d+e x} (c d-b e)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.30857, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}-\frac{2 e}{d \sqrt{d+e x} (c d-b e)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(b*x + c*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 37.8911, size = 87, normalized size = 0.85 \[ \frac{2 e}{d \sqrt{d + e x} \left (b e - c d\right )} + \frac{2 c^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b \left (b e - c d\right )^{\frac{3}{2}}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x),x)

[Out]

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

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Mathematica [A]  time = 0.337482, size = 102, normalized size = 1. \[ \frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b (c d-b e)^{3/2}}+\frac{2 e}{d \sqrt{d+e x} (b e-c d)}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.02, size = 97, normalized size = 1. \[ 2\,{\frac{{c}^{2}}{ \left ( be-cd \right ) b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{1}{b{d}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+2\,{\frac{e}{d \left ( be-cd \right ) \sqrt{ex+d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+b*x),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266188, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{e x + d} c d^{\frac{3}{2}} \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e - 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + 2 \, b \sqrt{d} e -{\left (c d - b e\right )} \sqrt{e x + d} \log \left (\frac{{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + d} d}{x}\right )}{{\left (b c d^{2} - b^{2} d e\right )} \sqrt{e x + d} \sqrt{d}}, \frac{2 \, \sqrt{e x + d} c d^{\frac{3}{2}} \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{-\frac{c}{c d - b e}}}{\sqrt{e x + d} c}\right ) - 2 \, b \sqrt{d} e +{\left (c d - b e\right )} \sqrt{e x + d} \log \left (\frac{{\left (e x + 2 \, d\right )} \sqrt{d} - 2 \, \sqrt{e x + d} d}{x}\right )}{{\left (b c d^{2} - b^{2} d e\right )} \sqrt{e x + d} \sqrt{d}}, -\frac{\sqrt{e x + d} c \sqrt{-d} d \sqrt{\frac{c}{c d - b e}} \log \left (\frac{c e x + 2 \, c d - b e - 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \sqrt{\frac{c}{c d - b e}}}{c x + b}\right ) + 2 \, b \sqrt{-d} e - 2 \,{\left (c d - b e\right )} \sqrt{e x + d} \arctan \left (\frac{d}{\sqrt{e x + d} \sqrt{-d}}\right )}{{\left (b c d^{2} - b^{2} d e\right )} \sqrt{e x + d} \sqrt{-d}}, \frac{2 \,{\left (\sqrt{e x + d} c \sqrt{-d} d \sqrt{-\frac{c}{c d - b e}} \arctan \left (-\frac{{\left (c d - b e\right )} \sqrt{-\frac{c}{c d - b e}}}{\sqrt{e x + d} c}\right ) - b \sqrt{-d} e +{\left (c d - b e\right )} \sqrt{e x + d} \arctan \left (\frac{d}{\sqrt{e x + d} \sqrt{-d}}\right )\right )}}{{\left (b c d^{2} - b^{2} d e\right )} \sqrt{e x + d} \sqrt{-d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (b + c x\right ) \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x),x)

[Out]

Integral(1/(x*(b + c*x)*(d + e*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.207289, size = 153, normalized size = 1.5 \[ -\frac{2 \, c^{2} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b c d - b^{2} e\right )} \sqrt{-c^{2} d + b c e}} - \frac{2 \, e}{{\left (c d^{2} - b d e\right )} \sqrt{x e + d}} + \frac{2 \, \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

-2*c^2*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b*c*d - b^2*e)*sqrt(-c^2*d
+ b*c*e)) - 2*e/((c*d^2 - b*d*e)*sqrt(x*e + d)) + 2*arctan(sqrt(x*e + d)/sqrt(-
d))/(b*sqrt(-d)*d)