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

Optimal. Leaf size=95 \[ -\frac{a e+c d x}{c \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{d^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]

[Out]

-((a*e + c*d*x)/(c*(c*d^2 + a*e^2)*Sqrt[a + c*x^2])) - (d^2*ArcTanh[(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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Rubi [A]  time = 0.180078, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{a e+c d x}{c \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{d^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((a*e + c*d*x)/(c*(c*d^2 + a*e^2)*Sqrt[a + c*x^2])) - (d^2*ArcTanh[(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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Rubi in Sympy [A]  time = 36.3881, size = 121, normalized size = 1.27 \[ - \frac{d^{2} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{1}{c e \sqrt{a + c x^{2}}} + \frac{d^{2} \left (a e + c d x\right )}{a e^{2} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} - \frac{d x}{a e^{2} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.248969, size = 121, normalized size = 1.27 \[ \frac{-a e-c d x}{c \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{d^2 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac{d^2 \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-(a*e) - c*d*x)/(c*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + (d^2*Log[d + e*x])/(c*d^2
+ a*e^2)^(3/2) - (d^2*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 = 0.014, size = 311, normalized size = 3.3 \[ -{\frac{1}{ce}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{dx}{a{e}^{2}}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{d}^{2}}{e \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{c{d}^{3}x}{{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{2}}{e \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.336172, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x + a e\right )} \sqrt{c x^{2} + a} -{\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \,{\left (a c^{2} d^{2} + a^{2} c e^{2} +{\left (c^{3} d^{2} + a c^{2} e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}}}, -\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x + a e\right )} \sqrt{c x^{2} + a} -{\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{{\left (a c^{2} d^{2} + a^{2} c e^{2} +{\left (c^{3} d^{2} + a c^{2} e^{2}\right )} x^{2}\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272045, size = 235, normalized size = 2.47 \[ -\frac{2 \, d^{2} \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} + a e^{2}\right )} \sqrt{-c d^{2} - a e^{2}}} - \frac{\frac{{\left (c^{2} d^{3} + a c d e^{2}\right )} x}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}} + \frac{a c d^{2} e + a^{2} e^{3}}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}{\sqrt{c x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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