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

Optimal. Leaf size=123 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e}+\frac{a (d-e x)}{c \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{d^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e \left (a e^2+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.329412, antiderivative size = 123, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e}+\frac{a (d-e x)}{c \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{d^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 44.6447, size = 167, normalized size = 1.36 \[ \frac{d^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} + \frac{d}{c e^{2} \sqrt{a + c x^{2}}} - \frac{x}{c e \sqrt{a + c x^{2}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{c^{\frac{3}{2}} e} - \frac{d^{3} \left (a e + c d x\right )}{a e^{3} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} + \frac{d^{2} x}{a e^{3} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.441252, size = 157, normalized size = 1.28 \[ \frac{\log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2} e}+\frac{a d-a e x}{c \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{d^3 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{e \left (a e^2+c d^2\right )^{3/2}}-\frac{d^3 \log (d+e x)}{e \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 354, normalized size = 2.9 \[{\frac{{d}^{2}x}{{e}^{3}a}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{x}{ce}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{1}{e}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{d}{{e}^{2}c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{{d}^{3}}{{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{4}cx}{{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{d}^{3}}{{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^2/e^3*x/a/(c*x^2+a)^(1/2)-1/e*x/c/(c*x^2+a)^(1/2)+1/e/c^(3/2)*ln(x*c^(1/2)+(c*
x^2+a)^(1/2))+d/e^2/c/(c*x^2+a)^(1/2)-d^3/e^2/(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^4/e^3/(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^3/e^2/(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/((c*x^2 + a)^(3/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.93785, 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/((c*x^2 + a)^(3/2)*(e*x + d)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.28136, size = 296, normalized size = 2.41 \[ -\frac{2 \, d^{3} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} e + a e^{3}\right )} \sqrt{-c d^{2} - a e^{2}}} - \frac{\frac{{\left (a c^{2} d^{2} e^{3} + a^{2} c e^{5}\right )} x}{c^{4} d^{4} e^{2} + 2 \, a c^{3} d^{2} e^{4} + a^{2} c^{2} e^{6}} - \frac{a c^{2} d^{3} e^{2} + a^{2} c d e^{4}}{c^{4} d^{4} e^{2} + 2 \, a c^{3} d^{2} e^{4} + a^{2} c^{2} e^{6}}}{\sqrt{c x^{2} + a}} - \frac{e^{\left (-1\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d^3*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2
))/((c*d^2*e + a*e^3)*sqrt(-c*d^2 - a*e^2)) - ((a*c^2*d^2*e^3 + a^2*c*e^5)*x/(c^
4*d^4*e^2 + 2*a*c^3*d^2*e^4 + a^2*c^2*e^6) - (a*c^2*d^3*e^2 + a^2*c*d*e^4)/(c^4*
d^4*e^2 + 2*a*c^3*d^2*e^4 + a^2*c^2*e^6))/sqrt(c*x^2 + a) - e^(-1)*ln(abs(-sqrt(
c)*x + sqrt(c*x^2 + a)))/c^(3/2)