3.392 \(\int \frac{x^5}{a c+b c x^3+d \sqrt{a+b x^3}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}-\frac{2 \left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^2 c^3}+\frac{x^3}{3 b c} \]

[Out]

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

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Rubi [A]  time = 0.367028, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{2 d \sqrt{a+b x^3}}{3 b^2 c^2}-\frac{2 \left (a c^2-d^2\right ) \log \left (c \sqrt{a+b x^3}+d\right )}{3 b^2 c^3}+\frac{x^3}{3 b c} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

2*Integral(x, (x, sqrt(a + b*x**3)))/(3*b**2*c) - 2*Integral(d, (x, sqrt(a + b*x
**3)))/(3*b**2*c**2) + 2*(-a*c**2 + d**2)*log(c*sqrt(a + b*x**3) + d)/(3*b**2*c*
*3)

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Mathematica [A]  time = 0.102777, size = 95, normalized size = 1.3 \[ \frac{\left (d^2-a c^2\right ) \log \left (a c^2+b c^2 x^3-d^2\right )+\left (2 d^2-2 a c^2\right ) \tanh ^{-1}\left (\frac{c \sqrt{a+b x^3}}{d}\right )+c \left (b c x^3-2 d \sqrt{a+b x^3}\right )}{3 b^2 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a*c + b*c*x^3 + d*Sqrt[a + b*x^3]),x]

[Out]

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

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Maple [C]  time = 0.021, size = 943, normalized size = 12.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(a*c+b*c*x^3+d*(b*x^3+a)^(1/2)),x)

[Out]

-2/3*d*(b*x^3+a)^(1/2)/b^2/c^2+1/3*I/b^4/d*2^(1/2)*sum((-a*b^2)^(1/3)*(1/2*I*b*(
2*x+1/b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1
/b*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*b
*(2*x+1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(b*x^
3+a)^(1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*b^2)^(2/3)*3^(1/2)+2*_alpha^
2*b^2-(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/b
*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),
-1/2*c^2/b*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*3^(1/2)*(-a*b^2)^(2/3)*_alph
a+I*3^(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-a*b^2)^(1/3)/(
-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3
*b*c^2+a*c^2-d^2))*a-1/3*I*d/b^4/c^2*2^(1/2)*sum((-a*b^2)^(1/3)*(1/2*I*b*(2*x+1/
b*((-a*b^2)^(1/3)-I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)*(b*(x-1/b*(-a
*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*b*(2*x+
1/b*((-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(
1/2)*(I*(-a*b^2)^(1/3)*3^(1/2)*_alpha*b-I*(-a*b^2)^(2/3)*3^(1/2)+2*_alpha^2*b^2-
(-a*b^2)^(1/3)*_alpha*b-(-a*b^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b
^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),-1/2*c
^2/b*(2*I*3^(1/2)*(-a*b^2)^(1/3)*_alpha^2*b-I*3^(1/2)*(-a*b^2)^(2/3)*_alpha+I*3^
(1/2)*a*b-3*(-a*b^2)^(2/3)*_alpha-3*a*b)/d^2,(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b
*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(_Z^3*b*c^2
+a*c^2-d^2))-1/3*a/c/b^2*ln(b*c^2*x^3+a*c^2-d^2)+1/3*x^3/b/c+1/3/b^2/c^3*d^2*ln(
b*c^2*x^3+a*c^2-d^2)

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Maxima [A]  time = 0.701437, size = 84, normalized size = 1.15 \[ \frac{\frac{{\left (b x^{3} + a\right )} c - 2 \, \sqrt{b x^{3} + a} d}{c^{2}} - \frac{2 \,{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right )}{c^{3}}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="maxima")

[Out]

1/3*(((b*x^3 + a)*c - 2*sqrt(b*x^3 + a)*d)/c^2 - 2*(a*c^2 - d^2)*log(sqrt(b*x^3
+ a)*c + d)/c^3)/b^2

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Fricas [A]  time = 0.279972, size = 159, normalized size = 2.18 \[ \frac{b c^{2} x^{3} - 2 \, \sqrt{b x^{3} + a} c d -{\left (a c^{2} - d^{2}\right )} \log \left (b c^{2} x^{3} + a c^{2} - d^{2}\right ) -{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c + d\right ) +{\left (a c^{2} - d^{2}\right )} \log \left (\sqrt{b x^{3} + a} c - d\right )}{3 \, b^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="fricas")

[Out]

1/3*(b*c^2*x^3 - 2*sqrt(b*x^3 + a)*c*d - (a*c^2 - d^2)*log(b*c^2*x^3 + a*c^2 - d
^2) - (a*c^2 - d^2)*log(sqrt(b*x^3 + a)*c + d) + (a*c^2 - d^2)*log(sqrt(b*x^3 +
a)*c - d))/(b^2*c^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(a*c+b*c*x**3+d*(b*x**3+a)**(1/2)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.271632, size = 97, normalized size = 1.33 \[ -\frac{\frac{2 \,{\left (a c^{2} - d^{2}\right )}{\rm ln}\left ({\left | \sqrt{b x^{3} + a} c + d \right |}\right )}{b c^{3}} - \frac{{\left (b x^{3} + a\right )} b c - 2 \, \sqrt{b x^{3} + a} b d}{b^{2} c^{2}}}{3 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*c*x^3 + a*c + sqrt(b*x^3 + a)*d),x, algorithm="giac")

[Out]

-1/3*(2*(a*c^2 - d^2)*ln(abs(sqrt(b*x^3 + a)*c + d))/(b*c^3) - ((b*x^3 + a)*b*c
- 2*sqrt(b*x^3 + a)*b*d)/(b^2*c^2))/b