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

Optimal. Leaf size=88 \[ \frac{d e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{d-e x}{\sqrt{a+c x^2} \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.142079, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{d e \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{d-e x}{\sqrt{a+c x^2} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.8604, size = 75, normalized size = 0.85 \[ \frac{d e \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{d - e x}{\sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.208528, size = 113, normalized size = 1.28 \[ \frac{e x-d}{\sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{d e \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{d e \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.018, size = 283, normalized size = 3.2 \[{\frac{x}{ae}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{d}{a{e}^{2}+c{d}^{2}}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{c{d}^{2}x}{e \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{d}{a{e}^{2}+c{d}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e*x/a/(c*x^2+a)^(1/2)-d/(a*e^2+c*d^2)/((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d^
2)/e^2)^(1/2)-d^2/e/(a*e^2+c*d^2)/a/((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+c*d^2)/e
^2)^(1/2)*c*x+d/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-
2*c*d/e*(x+d/e)+2*((a*e^2+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c-2*c*d/e*(x+d/e)+(a*e^2+
c*d^2)/e^2)^(1/2))/(x+d/e))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.340594, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a}{\left (e x - d\right )} +{\left (c d e x^{2} + a d e\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} - 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \,{\left (a c d^{2} + a^{2} e^{2} +{\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}}}, \frac{\sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a}{\left (e x - d\right )} -{\left (c d e x^{2} + a d e\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{{\left (a c d^{2} + a^{2} e^{2} +{\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.274818, size = 219, normalized size = 2.49 \[ \frac{2 \, d \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e}{{\left (c d^{2} + a e^{2}\right )} \sqrt{-c d^{2} - a e^{2}}} + \frac{\frac{{\left (c d^{2} e + a e^{3}\right )} x}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac{c d^{3} + a d e^{2}}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}}}{\sqrt{c x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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