3.345 \(\int \frac{x^2}{(d+e x)^2 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{d^2 \sqrt{a+c x^2}}{e (d+e x) \left (a e^2+c d^2\right )}+\frac{d \left (2 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c} e^2} \]

[Out]

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

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Rubi [A]  time = 0.349212, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{d^2 \sqrt{a+c x^2}}{e (d+e x) \left (a e^2+c d^2\right )}+\frac{d \left (2 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c} e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.8045, size = 167, normalized size = 1.22 \[ - \frac{c d^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{2} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{d^{2} \sqrt{a + c x^{2}}}{e \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} + \frac{2 d \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{2} \sqrt{a e^{2} + c d^{2}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.608454, size = 172, normalized size = 1.26 \[ \frac{d \left (\frac{\left (2 a e^2+c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{d e \sqrt{a+c x^2}}{(d+e x) \left (a e^2+c d^2\right )}\right )-\frac{\left (2 a d e^2+c d^3\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}+\frac{\log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}}}{e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 368, normalized size = 2.7 \[{\frac{1}{{e}^{2}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{d}^{2}}{{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ( x+{\frac{d}{e}} \right ) ^{-1}}-{\frac{c{d}^{3}}{{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+2\,{\frac{d}{{e}^{3}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.64733, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(2*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(c)*d^2*e - (c*d^3 + a*d*e^2 +
(c*d^2*e + a*e^3)*x)*sqrt(c*d^2 + a*e^2)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 +
a)*sqrt(c)) - (c*d^4 + 2*a*d^2*e^2 + (c*d^3*e + 2*a*d*e^3)*x)*sqrt(c)*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)))/((c*d^3*e^2 + a*d*e^4 + (c*d^2*e^3 + a*e^5)*x)*sqrt(c*d^2 + a*e
^2)*sqrt(c)), -1/2*(2*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(c)*d^2*e + 2*(c*
d^4 + 2*a*d^2*e^2 + (c*d^3*e + 2*a*d*e^3)*x)*sqrt(c)*arctan(sqrt(-c*d^2 - a*e^2)
*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))) - (c*d^3 + a*d*e^2 + (c*d^2*e
+ a*e^3)*x)*sqrt(-c*d^2 - a*e^2)*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt
(c)))/((c*d^3*e^2 + a*d*e^4 + (c*d^2*e^3 + a*e^5)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c
)), -1/2*(2*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(-c)*d^2*e - 2*(c*d^3 + a*d*
e^2 + (c*d^2*e + a*e^3)*x)*sqrt(c*d^2 + a*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)
) - (c*d^4 + 2*a*d^2*e^2 + (c*d^3*e + 2*a*d*e^3)*x)*sqrt(-c)*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)))/((c*d^3*e^2 + a*d*e^4 + (c*d^2*e^3 + a*e^5)*x)*sqrt(c*d^2 + a*e^2)*sqrt(-
c)), -(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-c)*d^2*e - (c*d^3 + a*d*e^2 +
(c*d^2*e + a*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (
c*d^4 + 2*a*d^2*e^2 + (c*d^3*e + 2*a*d*e^3)*x)*sqrt(-c)*arctan(sqrt(-c*d^2 - a*e
^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((c*d^3*e^2 + a*d*e^4 + (c
*d^2*e^3 + a*e^5)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{c x^{2} + a}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(sqrt(c*x^2 + a)*(e*x + d)^2), x)