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

Optimal. Leaf size=149 \[ \frac{\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c}}-\frac{\sqrt{d+e x} (2 c d-b e)}{b^2 (b+c x)}-\frac{d \sqrt{d+e x}}{b x (b+c x)} \]

[Out]

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

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Rubi [A]  time = 0.611917, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}-\frac{\sqrt{c d-b e} (4 c d-b e) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c}}-\frac{\sqrt{d+e x} (2 c d-b e)}{b^2 (b+c x)}-\frac{d \sqrt{d+e x}}{b x (b+c x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 63.4677, size = 129, normalized size = 0.87 \[ - \frac{d \sqrt{d + e x}}{b x \left (b + c x\right )} + \frac{\sqrt{d + e x} \left (b e - 2 c d\right )}{b^{2} \left (b + c x\right )} - \frac{\sqrt{d} \left (3 b e - 4 c d\right ) \operatorname{atanh}{\left (\frac{\sqrt{d + e x}}{\sqrt{d}} \right )}}{b^{3}} + \frac{\left (b e - 4 c d\right ) \sqrt{b e - c d} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d + e x}}{\sqrt{b e - c d}} \right )}}{b^{3} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.422972, size = 140, normalized size = 0.94 \[ \frac{-\frac{\left (b^2 e^2-5 b c d e+4 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{\sqrt{c} \sqrt{c d-b e}}+\frac{b \sqrt{d+e x} (-b d+b e x-2 c d x)}{x (b+c x)}+\sqrt{d} (4 c d-3 b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.027, size = 237, normalized size = 1.6 \[{\frac{{e}^{2}}{b \left ( cex+be \right ) }\sqrt{ex+d}}-{\frac{ced}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{{e}^{2}}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-5\,{\frac{ced}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{2}{d}^{2}}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{d}{{b}^{2}x}\sqrt{ex+d}}-3\,{\frac{e\sqrt{d}}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{{d}^{3/2}c}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)-e/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*c*d+e^2/b/((b*e-
c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))-5*e/b^2/((b*e-c*d)*c)^
(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*c*d+4/b^3/((b*e-c*d)*c)^(1/2)*
arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*c^2*d^2-d/b^2*(e*x+d)^(1/2)/x-3*e*d^
(1/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))+4*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/d^(
1/2))*c

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262167, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217756, size = 285, normalized size = 1.91 \[ \frac{{\left (4 \, c^{2} d^{2} - 5 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3}} - \frac{{\left (4 \, c d^{2} - 3 \, b d e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{2 \,{\left (x e + d\right )}^{\frac{3}{2}} c d e - 2 \, \sqrt{x e + d} c d^{2} e -{\left (x e + d\right )}^{\frac{3}{2}} b e^{2} + 2 \, \sqrt{x e + d} b d e^{2}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(
sqrt(-c^2*d + b*c*e)*b^3) - (4*c*d^2 - 3*b*d*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(
b^3*sqrt(-d)) - (2*(x*e + d)^(3/2)*c*d*e - 2*sqrt(x*e + d)*c*d^2*e - (x*e + d)^(
3/2)*b*e^2 + 2*sqrt(x*e + d)*b*d*e^2)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2
+ (x*e + d)*b*e - b*d*e)*b^2)