3.326 \(\int \frac{x^4}{(d+e x) \sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=195 \[ -\frac{d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2} e^4}+\frac{\sqrt{a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 c^2 e^3}-\frac{d^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4 \sqrt{a e^2+c d^2}}-\frac{7 d \sqrt{a+c x^2} (d+e x)}{6 c e^3}+\frac{\sqrt{a+c x^2} (d+e x)^2}{3 c e^3} \]

[Out]

((11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/(6*c^2*e^3) - (7*d*(d + e*x)*Sqrt[a + c*x
^2])/(6*c*e^3) + ((d + e*x)^2*Sqrt[a + c*x^2])/(3*c*e^3) - (d*(2*c*d^2 - a*e^2)*
ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(3/2)*e^4) - (d^4*ArcTanh[(a*e - c*d*
x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e^4*Sqrt[c*d^2 + a*e^2])

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Rubi [A]  time = 0.833369, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2} e^4}+\frac{\sqrt{a+c x^2} \left (11 c d^2-4 a e^2\right )}{6 c^2 e^3}-\frac{d^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4 \sqrt{a e^2+c d^2}}-\frac{7 d \sqrt{a+c x^2} (d+e x)}{6 c e^3}+\frac{\sqrt{a+c x^2} (d+e x)^2}{3 c e^3} \]

Antiderivative was successfully verified.

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

[Out]

((11*c*d^2 - 4*a*e^2)*Sqrt[a + c*x^2])/(6*c^2*e^3) - (7*d*(d + e*x)*Sqrt[a + c*x
^2])/(6*c*e^3) + ((d + e*x)^2*Sqrt[a + c*x^2])/(3*c*e^3) - (d*(2*c*d^2 - a*e^2)*
ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(3/2)*e^4) - (d^4*ArcTanh[(a*e - c*d*
x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e^4*Sqrt[c*d^2 + a*e^2])

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Rubi in Sympy [A]  time = 38.4035, size = 189, normalized size = 0.97 \[ - \frac{a \sqrt{a + c x^{2}}}{c^{2} e} + \frac{a d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 c^{\frac{3}{2}} e^{2}} - \frac{d^{4} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{4} \sqrt{a e^{2} + c d^{2}}} + \frac{d^{2} \sqrt{a + c x^{2}}}{c e^{3}} - \frac{d x \sqrt{a + c x^{2}}}{2 c e^{2}} + \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c^{2} e} - \frac{d^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(e*x+d)/(c*x**2+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.798338, size = 178, normalized size = 0.91 \[ \frac{-\frac{3 d \left (2 c d^2-a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}}+\frac{e \sqrt{a+c x^2} \left (c \left (6 d^2-3 d e x+2 e^2 x^2\right )-4 a e^2\right )}{c^2}-\frac{6 d^4 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}+\frac{6 d^4 \log (d+e x)}{\sqrt{a e^2+c d^2}}}{6 e^4} \]

Antiderivative was successfully verified.

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

[Out]

((e*Sqrt[a + c*x^2]*(-4*a*e^2 + c*(6*d^2 - 3*d*e*x + 2*e^2*x^2)))/c^2 + (6*d^4*L
og[d + e*x])/Sqrt[c*d^2 + a*e^2] - (3*d*(2*c*d^2 - a*e^2)*Log[c*x + Sqrt[c]*Sqrt
[a + c*x^2]])/c^(3/2) - (6*d^4*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])/(6*e^4)

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Maple [A]  time = 0.021, size = 260, normalized size = 1.3 \[{\frac{{x}^{2}}{3\,ce}\sqrt{c{x}^{2}+a}}-{\frac{2\,a}{3\,e{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{{d}^{2}}{{e}^{3}c}\sqrt{c{x}^{2}+a}}-{\frac{{d}^{4}}{{e}^{5}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{{d}^{3}}{{e}^{4}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{dx}{2\,{e}^{2}c}\sqrt{c{x}^{2}+a}}+{\frac{ad}{2\,{e}^{2}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(e*x+d)/(c*x^2+a)^(1/2),x)

[Out]

1/3/e*x^2/c*(c*x^2+a)^(1/2)-2/3/e*a/c^2*(c*x^2+a)^(1/2)+d^2/e^3/c*(c*x^2+a)^(1/2
)-d^4/e^5/((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*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-1/2*d/e^2*x/c*(c*x^2+a)^(1
/2)+1/2*d/e^2*a/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/(sqrt(c*x^2 + a)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.92123, 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/(sqrt(c*x^2 + a)*(e*x + d)),x, algorithm="fricas")

[Out]

[1/12*(6*c^(5/2)*d^4*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*(2*c*e^3*x^2 - 3*c*d*e^2*x + 6
*c*d^2*e - 4*a*e^3)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(c) - 3*(2*c^2*d^3 -
a*c*d*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^(5/2)*e^4), 1/12*(12*c^(5/2)*d^4*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*(2*c*e^3*x^2 - 3*c
*d*e^2*x + 6*c*d^2*e - 4*a*e^3)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(c) - 3
*(2*c^2*d^3 - a*c*d*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^(5/2)*e^4), 1/6*(3*sqrt(-c)*c^2*d^4*l
og(((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*c*e^3*x^2 - 3*c*d*e^2*x + 6*c*d^2*e - 4*a*e^3)*sqrt
(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(-c) - 3*(2*c^2*d^3 - a*c*d*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^2*e
^4), 1/6*(6*sqrt(-c)*c^2*d^4*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*c*e^3*x^2 - 3*c*d*e^2*x + 6*c*d^2*e - 4*a*e^3)*sq
rt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-c) - 3*(2*c^2*d^3 - a*c*d*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^2*e^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(e*x+d)/(c*x**2+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.278003, size = 220, normalized size = 1.13 \[ \frac{2 \, d^{4} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-4\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \frac{1}{6} \, \sqrt{c x^{2} + a}{\left (x{\left (\frac{2 \, x e^{\left (-1\right )}}{c} - \frac{3 \, d e^{\left (-2\right )}}{c}\right )} + \frac{2 \,{\left (3 \, c^{2} d^{2} e^{7} - 2 \, a c e^{9}\right )} e^{\left (-10\right )}}{c^{3}}\right )} + \frac{{\left (2 \, c^{\frac{3}{2}} d^{3} - a \sqrt{c} d e^{2}\right )} e^{\left (-4\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(c*x^2 + a)*(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)
)*e^(-4)/sqrt(-c*d^2 - a*e^2) + 1/6*sqrt(c*x^2 + a)*(x*(2*x*e^(-1)/c - 3*d*e^(-2
)/c) + 2*(3*c^2*d^2*e^7 - 2*a*c*e^9)*e^(-10)/c^3) + 1/2*(2*c^(3/2)*d^3 - a*sqrt(
c)*d*e^2)*e^(-4)*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^2