3.334 \(\int \frac{x^4}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=146 \[ -\frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e^2}+\frac{a (a e+c d x)}{c^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{\sqrt{a+c x^2}}{c^2 e}-\frac{d^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}} \]

[Out]

(a*(a*e + c*d*x))/(c^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + Sqrt[a + c*x^2]/(c^2*e
) - (d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(c^(3/2)*e^2) - (d^4*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 = 0.569879, antiderivative size = 146, 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 \[ -\frac{d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{c^{3/2} e^2}+\frac{a (a e+c d x)}{c^2 \sqrt{a+c x^2} \left (a e^2+c d^2\right )}+\frac{\sqrt{a+c x^2}}{c^2 e}-\frac{d^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((d + e*x)*(a + c*x^2)^(3/2)),x]

[Out]

(a*(a*e + c*d*x))/(c^2*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) + Sqrt[a + c*x^2]/(c^2*e
) - (d*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(c^(3/2)*e^2) - (d^4*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 = 54.9717, size = 209, normalized size = 1.43 \[ \frac{a}{c^{2} e \sqrt{a + c x^{2}}} - \frac{d^{4} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{2} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{d^{2}}{c e^{3} \sqrt{a + c x^{2}}} + \frac{d x}{c e^{2} \sqrt{a + c x^{2}}} + \frac{\sqrt{a + c x^{2}}}{c^{2} e} - \frac{d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{c^{\frac{3}{2}} e^{2}} + \frac{d^{4} \left (a e + c d x\right )}{a e^{4} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} - \frac{d^{3} x}{a e^{4} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(e*x+d)/(c*x**2+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.503546, size = 173, normalized size = 1.18 \[ -\frac{d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2} e^2}+\frac{\sqrt{a+c x^2} \left (\frac{a (a e+c d x)}{\left (a+c x^2\right ) \left (a e^2+c d^2\right )}+\frac{1}{e}\right )}{c^2}-\frac{d^4 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{e^2 \left (a e^2+c d^2\right )^{3/2}}+\frac{d^4 \log (d+e x)}{e^2 \left (a e^2+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((d + e*x)*(a + c*x^2)^(3/2)),x]

[Out]

(Sqrt[a + c*x^2]*(e^(-1) + (a*(a*e + c*d*x))/((c*d^2 + a*e^2)*(a + c*x^2))))/c^2
+ (d^4*Log[d + e*x])/(e^2*(c*d^2 + a*e^2)^(3/2)) - (d*Log[c*x + Sqrt[c]*Sqrt[a
+ c*x^2]])/(c^(3/2)*e^2) - (d^4*Log[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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Maple [B]  time = 0.024, size = 396, normalized size = 2.7 \[{\frac{{x}^{2}}{ce}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+2\,{\frac{a}{e{c}^{2}\sqrt{c{x}^{2}+a}}}-{\frac{{d}^{2}}{{e}^{3}c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{{d}^{4}}{{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{{d}^{5}cx}{{e}^{4} \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{4}}{{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{3}x}{{e}^{4}a}{\frac{1}{\sqrt{c{x}^{2}+a}}}}+{\frac{dx}{{e}^{2}c}{\frac{1}{\sqrt{c{x}^{2}+a}}}}-{\frac{d}{{e}^{2}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(e*x+d)/(c*x^2+a)^(3/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.06737, 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^4/((c*x^2 + a)^(3/2)*(e*x + d)),x, algorithm="fricas")

[Out]

[1/2*(2*(a*c*d*e^2*x + a*c*d^2*e + 2*a^2*e^3 + (c^2*d^2*e + a*c*e^3)*x^2)*sqrt(c
*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(c) + (a*c^2*d^3 + a^2*c*d*e^2 + (c^3*d^3 + a*
c^2*d*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)*sq
rt(c)) + (c^3*d^4*x^2 + a*c^2*d^4)*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^3*d^2
*e^2 + a^2*c^2*e^4 + (c^4*d^2*e^2 + a*c^3*e^4)*x^2)*sqrt(c*d^2 + a*e^2)*sqrt(c))
, 1/2*(2*(a*c*d*e^2*x + a*c*d^2*e + 2*a^2*e^3 + (c^2*d^2*e + a*c*e^3)*x^2)*sqrt(
-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(c) + 2*(c^3*d^4*x^2 + a*c^2*d^4)*sqrt(c)*ar
ctan(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^3 + a^2*c*d*e^2 + (c^3*d^3 + a*c^2*d*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^3*d^2*e^2 + a^2*c^2*e^4 + (c
^4*d^2*e^2 + a*c^3*e^4)*x^2)*sqrt(-c*d^2 - a*e^2)*sqrt(c)), 1/2*(2*(a*c*d*e^2*x
+ a*c*d^2*e + 2*a^2*e^3 + (c^2*d^2*e + a*c*e^3)*x^2)*sqrt(c*d^2 + a*e^2)*sqrt(c*
x^2 + a)*sqrt(-c) - 2*(a*c^2*d^3 + a^2*c*d*e^2 + (c^3*d^3 + a*c^2*d*e^2)*x^2)*sq
rt(c*d^2 + a*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (c^3*d^4*x^2 + a*c^2*d^4)
*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)*s
qrt(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^3*d^2*e^2 + a^2*c^2*e^4 + (c^4*d^2*e^2
+ a*c^3*e^4)*x^2)*sqrt(c*d^2 + a*e^2)*sqrt(-c)), ((a*c*d*e^2*x + a*c*d^2*e + 2*
a^2*e^3 + (c^2*d^2*e + a*c*e^3)*x^2)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-
c) - (a*c^2*d^3 + a^2*c*d*e^2 + (c^3*d^3 + a*c^2*d*e^2)*x^2)*sqrt(-c*d^2 - a*e^2
)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (c^3*d^4*x^2 + a*c^2*d^4)*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^3*
d^2*e^2 + a^2*c^2*e^4 + (c^4*d^2*e^2 + a*c^3*e^4)*x^2)*sqrt(-c*d^2 - a*e^2)*sqrt
(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(e*x+d)/(c*x**2+a)**(3/2),x)

[Out]

Integral(x**4/((a + c*x**2)**(3/2)*(d + e*x)), x)

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GIAC/XCAS [A]  time = 0.281573, size = 404, normalized size = 2.77 \[ \frac{2 \, d^{4} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right )}{{\left (c d^{2} e^{2} + a e^{4}\right )} \sqrt{-c d^{2} - a e^{2}}} + \frac{d e^{\left (-2\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{c^{\frac{3}{2}}} + \frac{{\left (\frac{{\left (c^{4} d^{4} e^{5} + 2 \, a c^{3} d^{2} e^{7} + a^{2} c^{2} e^{9}\right )} x}{c^{5} d^{4} e^{6} + 2 \, a c^{4} d^{2} e^{8} + a^{2} c^{3} e^{10}} + \frac{a c^{3} d^{3} e^{6} + a^{2} c^{2} d e^{8}}{c^{5} d^{4} e^{6} + 2 \, a c^{4} d^{2} e^{8} + a^{2} c^{3} e^{10}}\right )} x + \frac{a c^{3} d^{4} e^{5} + 3 \, a^{2} c^{2} d^{2} e^{7} + 2 \, a^{3} c e^{9}}{c^{5} d^{4} e^{6} + 2 \, a c^{4} d^{2} e^{8} + a^{2} c^{3} e^{10}}}{\sqrt{c x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((c*x^2 + a)^(3/2)*(e*x + d)),x, algorithm="giac")

[Out]

2*d^4*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a*e^2)
)/((c*d^2*e^2 + a*e^4)*sqrt(-c*d^2 - a*e^2)) + d*e^(-2)*ln(abs(-sqrt(c)*x + sqrt
(c*x^2 + a)))/c^(3/2) + (((c^4*d^4*e^5 + 2*a*c^3*d^2*e^7 + a^2*c^2*e^9)*x/(c^5*d
^4*e^6 + 2*a*c^4*d^2*e^8 + a^2*c^3*e^10) + (a*c^3*d^3*e^6 + a^2*c^2*d*e^8)/(c^5*
d^4*e^6 + 2*a*c^4*d^2*e^8 + a^2*c^3*e^10))*x + (a*c^3*d^4*e^5 + 3*a^2*c^2*d^2*e^
7 + 2*a^3*c*e^9)/(c^5*d^4*e^6 + 2*a*c^4*d^2*e^8 + a^2*c^3*e^10))/sqrt(c*x^2 + a)