3.344 \(\int \frac{x^3}{(d+e x)^2 \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{d^2 \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^3 \left (a e^2+c d^2\right )^{3/2}}+\frac{d^3 \sqrt{a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}-\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c} e^3}+\frac{\sqrt{a+c x^2}}{c e^2} \]

[Out]

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

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Rubi [A]  time = 0.595137, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{d^2 \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^3 \left (a e^2+c d^2\right )^{3/2}}+\frac{d^3 \sqrt{a+c x^2}}{e^2 (d+e x) \left (a e^2+c d^2\right )}-\frac{2 d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{\sqrt{c} e^3}+\frac{\sqrt{a+c x^2}}{c e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.5456, size = 189, normalized size = 1.18 \[ \frac{c d^{4} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{3} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} + \frac{d^{3} \sqrt{a + c x^{2}}}{e^{2} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} - \frac{3 d^{2} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{3} \sqrt{a e^{2} + c d^{2}}} + \frac{\sqrt{a + c x^{2}}}{c e^{2}} - \frac{2 d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*d**4*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(e**3*(a*e*
*2 + c*d**2)**(3/2)) + d**3*sqrt(a + c*x**2)/(e**2*(d + e*x)*(a*e**2 + c*d**2))
- 3*d**2*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(e**3*sqr
t(a*e**2 + c*d**2)) + sqrt(a + c*x**2)/(c*e**2) - 2*d*atanh(sqrt(c)*x/sqrt(a + c
*x**2))/(sqrt(c)*e**3)

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Mathematica [A]  time = 0.470515, size = 184, normalized size = 1.15 \[ \frac{-\frac{d^2 \left (3 a e^2+2 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}+\frac{d^2 \left (3 a e^2+2 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}+e \sqrt{a+c x^2} \left (\frac{d^3}{(d+e x) \left (a e^2+c d^2\right )}+\frac{1}{c}\right )-\frac{2 d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{\sqrt{c}}}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.016, size = 386, normalized size = 2.4 \[{\frac{1}{c{e}^{2}}\sqrt{c{x}^{2}+a}}-2\,{\frac{d\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) }{{e}^{3}\sqrt{c}}}-3\,{\frac{{d}^{2}}{{e}^{4}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{d}^{3}}{{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ( x+{\frac{d}{e}} \right ) ^{-1}}+{\frac{{d}^{4}c}{{e}^{4} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2+a)^(1/2)/c/e^2-2/e^3*d*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-3/e^4*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))+d^3/
e^3/(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^4/e^4*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))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.6053, 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^3/(sqrt(c*x^2 + a)*(e*x + d)^2),x, algorithm="fricas")

[Out]

[1/2*(2*(2*c*d^3*e + a*d*e^3 + (c*d^2*e^2 + a*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c
*x^2 + a)*sqrt(c) + 2*(c^2*d^4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*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)) + (2*c^2*d^5 +
3*a*c*d^3*e^2 + (2*c^2*d^4*e + 3*a*c*d^2*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^2*d^3*e^3 + a*c*d*e^5 + (c^2*d^2*e^4 + a*c*e^6)*x)*sqrt(c*d^2 + a*e^2)*sq
rt(c)), ((2*c*d^3*e + a*d*e^3 + (c*d^2*e^2 + a*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt
(c*x^2 + a)*sqrt(c) + (2*c^2*d^5 + 3*a*c*d^3*e^2 + (2*c^2*d^4*e + 3*a*c*d^2*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^2*d^4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*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^2*d^3*e^3 + a*c*d*e^
5 + (c^2*d^2*e^4 + a*c*e^6)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c)), 1/2*(2*(2*c*d^3*e
+ a*d*e^3 + (c*d^2*e^2 + a*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(-c)
- 4*(c^2*d^4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*arct
an(sqrt(-c)*x/sqrt(c*x^2 + a)) + (2*c^2*d^5 + 3*a*c*d^3*e^2 + (2*c^2*d^4*e + 3*a
*c*d^2*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^2*d^3*e^3 + a*c*d*e^5 + (c
^2*d^2*e^4 + a*c*e^6)*x)*sqrt(c*d^2 + a*e^2)*sqrt(-c)), ((2*c*d^3*e + a*d*e^3 +
(c*d^2*e^2 + a*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-c) - 2*(c^2*d^
4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*sqrt(-c*d^2 - a*e^2)*arctan(sqrt(-c
)*x/sqrt(c*x^2 + a)) + (2*c^2*d^5 + 3*a*c*d^3*e^2 + (2*c^2*d^4*e + 3*a*c*d^2*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^2*d^3*e^3 + a*c*d*e^5 + (c^2*d^2*e^4 + a*c*e^6)*x)*sqrt(-c*d^2 -
a*e^2)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/(sqrt(c*x^2 + a)*(e*x + d)^2), x)