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

Optimal. Leaf size=204 \[ \frac{\left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2} e^4}-\frac{d^4 \sqrt{a+c x^2}}{e^3 (d+e x) \left (a e^2+c d^2\right )}+\frac{d^3 \left (4 a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4 \left (a e^2+c d^2\right )^{3/2}}-\frac{5 d \sqrt{a+c x^2}}{2 c e^3}+\frac{\sqrt{a+c x^2} (d+e x)}{2 c e^3} \]

[Out]

(-5*d*Sqrt[a + c*x^2])/(2*c*e^3) - (d^4*Sqrt[a + c*x^2])/(e^3*(c*d^2 + a*e^2)*(d
+ e*x)) + ((d + e*x)*Sqrt[a + c*x^2])/(2*c*e^3) + ((6*c*d^2 - a*e^2)*ArcTanh[(S
qrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(3/2)*e^4) + (d^3*(3*c*d^2 + 4*a*e^2)*ArcTanh[(
a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e^4*(c*d^2 + a*e^2)^(3/2))

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Rubi [A]  time = 0.906049, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2} e^4}-\frac{d^4 \sqrt{a+c x^2}}{e^3 (d+e x) \left (a e^2+c d^2\right )}+\frac{d^3 \left (4 a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4 \left (a e^2+c d^2\right )^{3/2}}-\frac{5 d \sqrt{a+c x^2}}{2 c e^3}+\frac{\sqrt{a+c x^2} (d+e x)}{2 c e^3} \]

Antiderivative was successfully verified.

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

[Out]

(-5*d*Sqrt[a + c*x^2])/(2*c*e^3) - (d^4*Sqrt[a + c*x^2])/(e^3*(c*d^2 + a*e^2)*(d
+ e*x)) + ((d + e*x)*Sqrt[a + c*x^2])/(2*c*e^3) + ((6*c*d^2 - a*e^2)*ArcTanh[(S
qrt[c]*x)/Sqrt[a + c*x^2]])/(2*c^(3/2)*e^4) + (d^3*(3*c*d^2 + 4*a*e^2)*ArcTanh[(
a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(e^4*(c*d^2 + a*e^2)^(3/2))

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Rubi in Sympy [A]  time = 44.7607, size = 243, normalized size = 1.19 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 c^{\frac{3}{2}} e^{2}} - \frac{c d^{5} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{4} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{d^{4} \sqrt{a + c x^{2}}}{e^{3} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} + \frac{4 d^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{4} \sqrt{a e^{2} + c d^{2}}} - \frac{2 d \sqrt{a + c x^{2}}}{c e^{3}} + \frac{x \sqrt{a + c x^{2}}}{2 c e^{2}} + \frac{3 d^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{\sqrt{c} e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(2*c**(3/2)*e**2) - c*d**5*atanh((a*e - c*d
*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(e**4*(a*e**2 + c*d**2)**(3/2)) -
d**4*sqrt(a + c*x**2)/(e**3*(d + e*x)*(a*e**2 + c*d**2)) + 4*d**3*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)) - 2
*d*sqrt(a + c*x**2)/(c*e**3) + x*sqrt(a + c*x**2)/(2*c*e**2) + 3*d**2*atanh(sqrt
(c)*x/sqrt(a + c*x**2))/(sqrt(c)*e**4)

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Mathematica [A]  time = 0.636178, size = 208, normalized size = 1.02 \[ \frac{\frac{\left (6 c d^2-a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{c^{3/2}}+e \sqrt{a+c x^2} \left (\frac{e x-4 d}{c}-\frac{2 d^4}{(d+e x) \left (a e^2+c d^2\right )}\right )+\frac{2 d^3 \left (4 a e^2+3 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{2 d^3 \left (4 a e^2+3 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}}{2 e^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.018, size = 435, normalized size = 2.1 \[{\frac{x}{2\,c{e}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{a}{2\,{e}^{2}}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{d}^{4}}{{e}^{4} \left ( a{e}^{2}+c{d}^{2} \right ) }\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \left ( x+{\frac{d}{e}} \right ) ^{-1}}-{\frac{{d}^{5}c}{{e}^{5} \left ( a{e}^{2}+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+3\,{\frac{{d}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) }{{e}^{4}\sqrt{c}}}-2\,{\frac{d\sqrt{c{x}^{2}+a}}{c{e}^{3}}}+4\,{\frac{{d}^{3}}{{e}^{5}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c-2\,{\frac{cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 66.849, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(2*(6*c*d^4*e + 4*a*d^2*e^3 - (c*d^2*e^3 + a*e^5)*x^2 + 3*(c*d^3*e^2 + a*d
*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)*sqrt(c) + (6*c^2*d^5 + 5*a*c*d^3*e^
2 - a^2*d*e^4 + (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5)*x)*sqrt(c*d^2 + a*e^2)*l
og(2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) - 2*(3*c^2*d^6 + 4*a*c*d^4*e^2
+ (3*c^2*d^5*e + 4*a*c*d^3*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^4 + a*c*d*e^6 + (c^2*d^2*e^5 + a*c*e^7)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c)), -1/4*
(2*(6*c*d^4*e + 4*a*d^2*e^3 - (c*d^2*e^3 + a*e^5)*x^2 + 3*(c*d^3*e^2 + a*d*e^4)*
x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(c) + 4*(3*c^2*d^6 + 4*a*c*d^4*e^2 +
(3*c^2*d^5*e + 4*a*c*d^3*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))) + (6*c^2*d^5 + 5*a*c*d^3*e^2 - a^2*d*e^4
+ (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5)*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^4 + a*c*d*e^6 + (c^2*d^2*e^5 +
a*c*e^7)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c)), -1/2*((6*c*d^4*e + 4*a*d^2*e^3 - (c*
d^2*e^3 + a*e^5)*x^2 + 3*(c*d^3*e^2 + a*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2
+ a)*sqrt(-c) - (6*c^2*d^5 + 5*a*c*d^3*e^2 - a^2*d*e^4 + (6*c^2*d^4*e + 5*a*c*d
^2*e^3 - a^2*e^5)*x)*sqrt(c*d^2 + a*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (3
*c^2*d^6 + 4*a*c*d^4*e^2 + (3*c^2*d^5*e + 4*a*c*d^3*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^4 + a*c*d*e^6 + (c^2*d^2*e^5 + a*c*e^7)*x)*sqrt(c*d^2
+ a*e^2)*sqrt(-c)), -1/2*((6*c*d^4*e + 4*a*d^2*e^3 - (c*d^2*e^3 + a*e^5)*x^2 +
3*(c*d^3*e^2 + a*d*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a)*sqrt(-c) - (6*c^
2*d^5 + 5*a*c*d^3*e^2 - a^2*d*e^4 + (6*c^2*d^4*e + 5*a*c*d^2*e^3 - a^2*e^5)*x)*s
qrt(-c*d^2 - a*e^2)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + 2*(3*c^2*d^6 + 4*a*c*d^
4*e^2 + (3*c^2*d^5*e + 4*a*c*d^3*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^4 + a*c*d*e^6 + (c^2
*d^2*e^5 + a*c*e^7)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(-c))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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