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

Optimal. Leaf size=152 \[ \frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2} e^3}+\frac{d^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^3 \sqrt{a e^2+c d^2}}-\frac{3 d \sqrt{a+c x^2}}{2 c e^2}+\frac{\sqrt{a+c x^2} (d+e x)}{2 c e^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.503231, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2} e^3}+\frac{d^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^3 \sqrt{a e^2+c d^2}}-\frac{3 d \sqrt{a+c x^2}}{2 c e^2}+\frac{\sqrt{a+c x^2} (d+e x)}{2 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.2701, size = 146, normalized size = 0.96 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 c^{\frac{3}{2}} e} + \frac{d^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{3} \sqrt{a e^{2} + c d^{2}}} - \frac{d \sqrt{a + c x^{2}}}{c e^{2}} + \frac{x \sqrt{a + c x^{2}}}{2 c e} + \frac{d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c} e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.402967, size = 155, normalized size = 1.02 \[ \frac{\frac{\left (2 c d^2-a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}}+\frac{2 d^3 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}-\frac{2 d^3 \log (d+e x)}{\sqrt{a e^2+c d^2}}+\frac{e \sqrt{a+c x^2} (e x-2 d)}{c}}{2 e^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 217, normalized size = 1.4 \[{\frac{{d}^{2}}{{e}^{3}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{x}{2\,ce}\sqrt{c{x}^{2}+a}}-{\frac{a}{2\,e}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{d}{c{e}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{{d}^{3}}{{e}^{4}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 2.9921, 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)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.281944, size = 174, normalized size = 1.14 \[ -\frac{2 \, d^{3} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-3\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \frac{1}{2} \, \sqrt{c x^{2} + a}{\left (\frac{x e^{\left (-1\right )}}{c} - \frac{2 \, d e^{\left (-2\right )}}{c}\right )} - \frac{{\left (2 \, c d^{2} - a e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(c*x^2 + a)*(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
))*e^(-3)/sqrt(-c*d^2 - a*e^2) + 1/2*sqrt(c*x^2 + a)*(x*e^(-1)/c - 2*d*e^(-2)/c)
- 1/2*(2*c*d^2 - a*e^2)*e^(-3)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(3/2)