3.100 \(\int f^{a+b x^3} x^5 \, dx\)

Optimal. Leaf size=44 \[ \frac{x^3 f^{a+b x^3}}{3 b \log (f)}-\frac{f^{a+b x^3}}{3 b^2 \log ^2(f)} \]

[Out]

-f^(a + b*x^3)/(3*b^2*Log[f]^2) + (f^(a + b*x^3)*x^3)/(3*b*Log[f])

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Rubi [A]  time = 0.0739836, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^3 f^{a+b x^3}}{3 b \log (f)}-\frac{f^{a+b x^3}}{3 b^2 \log ^2(f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^3)*x^5,x]

[Out]

-f^(a + b*x^3)/(3*b^2*Log[f]^2) + (f^(a + b*x^3)*x^3)/(3*b*Log[f])

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Rubi in Sympy [A]  time = 6.54497, size = 36, normalized size = 0.82 \[ \frac{f^{a + b x^{3}} x^{3}}{3 b \log{\left (f \right )}} - \frac{f^{a + b x^{3}}}{3 b^{2} \log{\left (f \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(b*x**3+a)*x**5,x)

[Out]

f**(a + b*x**3)*x**3/(3*b*log(f)) - f**(a + b*x**3)/(3*b**2*log(f)**2)

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Mathematica [A]  time = 0.0108529, size = 29, normalized size = 0.66 \[ \frac{f^{a+b x^3} \left (b x^3 \log (f)-1\right )}{3 b^2 \log ^2(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^3)*x^5,x]

[Out]

(f^(a + b*x^3)*(-1 + b*x^3*Log[f]))/(3*b^2*Log[f]^2)

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Maple [A]  time = 0.007, size = 28, normalized size = 0.6 \[{\frac{ \left ( b{x}^{3}\ln \left ( f \right ) -1 \right ){f}^{b{x}^{3}+a}}{3\, \left ( \ln \left ( f \right ) \right ) ^{2}{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(b*x^3+a)*x^5,x)

[Out]

1/3*(b*x^3*ln(f)-1)*f^(b*x^3+a)/ln(f)^2/b^2

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Maxima [A]  time = 0.829769, size = 43, normalized size = 0.98 \[ \frac{{\left (b f^{a} x^{3} \log \left (f\right ) - f^{a}\right )} f^{b x^{3}}}{3 \, b^{2} \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^5,x, algorithm="maxima")

[Out]

1/3*(b*f^a*x^3*log(f) - f^a)*f^(b*x^3)/(b^2*log(f)^2)

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Fricas [A]  time = 0.317973, size = 36, normalized size = 0.82 \[ \frac{{\left (b x^{3} \log \left (f\right ) - 1\right )} f^{b x^{3} + a}}{3 \, b^{2} \log \left (f\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^5,x, algorithm="fricas")

[Out]

1/3*(b*x^3*log(f) - 1)*f^(b*x^3 + a)/(b^2*log(f)^2)

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Sympy [A]  time = 0.220419, size = 41, normalized size = 0.93 \[ \begin{cases} \frac{f^{a + b x^{3}} \left (b x^{3} \log{\left (f \right )} - 1\right )}{3 b^{2} \log{\left (f \right )}^{2}} & \text{for}\: 3 b^{2} \log{\left (f \right )}^{2} \neq 0 \\\frac{x^{6}}{6} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(b*x**3+a)*x**5,x)

[Out]

Piecewise((f**(a + b*x**3)*(b*x**3*log(f) - 1)/(3*b**2*log(f)**2), Ne(3*b**2*log
(f)**2, 0)), (x**6/6, True))

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GIAC/XCAS [A]  time = 0.259642, size = 932, normalized size = 21.18 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^3 + a)*x^5,x, algorithm="giac")

[Out]

1/3*(2*((b*x^3*ln(abs(f)) - 1)*(pi^2*b^2*sign(f) - pi^2*b^2 + 2*b^2*ln(abs(f))^2
)/((pi^2*b^2*sign(f) - pi^2*b^2 + 2*b^2*ln(abs(f))^2)^2 + 4*(pi*b^2*ln(abs(f))*s
ign(f) - pi*b^2*ln(abs(f)))^2) + (pi*b*x^3*sign(f) - pi*b*x^3)*(pi*b^2*ln(abs(f)
)*sign(f) - pi*b^2*ln(abs(f)))/((pi^2*b^2*sign(f) - pi^2*b^2 + 2*b^2*ln(abs(f))^
2)^2 + 4*(pi*b^2*ln(abs(f))*sign(f) - pi*b^2*ln(abs(f)))^2))*cos(-1/2*pi*b*x^3*s
ign(f) + 1/2*pi*b*x^3 - 1/2*pi*a*sign(f) + 1/2*pi*a) + ((pi*b*x^3*sign(f) - pi*b
*x^3)*(pi^2*b^2*sign(f) - pi^2*b^2 + 2*b^2*ln(abs(f))^2)/((pi^2*b^2*sign(f) - pi
^2*b^2 + 2*b^2*ln(abs(f))^2)^2 + 4*(pi*b^2*ln(abs(f))*sign(f) - pi*b^2*ln(abs(f)
))^2) - 4*(b*x^3*ln(abs(f)) - 1)*(pi*b^2*ln(abs(f))*sign(f) - pi*b^2*ln(abs(f)))
/((pi^2*b^2*sign(f) - pi^2*b^2 + 2*b^2*ln(abs(f))^2)^2 + 4*(pi*b^2*ln(abs(f))*si
gn(f) - pi*b^2*ln(abs(f)))^2))*sin(-1/2*pi*b*x^3*sign(f) + 1/2*pi*b*x^3 - 1/2*pi
*a*sign(f) + 1/2*pi*a))*e^(b*x^3*ln(abs(f)) + a*ln(abs(f))) - 1/6*((2*b*i*x^3*ln
(abs(f)) - pi*b*x^3*sign(f) + pi*b*x^3 - 2*i)*e^(1/2*(pi*b*x^3*(sign(f) - 1) + p
i*a*(sign(f) - 1))*i)/(2*pi*b^2*i*ln(abs(f))*sign(f) - 2*pi*b^2*i*ln(abs(f)) + p
i^2*b^2*sign(f) - pi^2*b^2 + 2*b^2*ln(abs(f))^2) + (2*b*i*x^3*ln(abs(f)) + pi*b*
x^3*sign(f) - pi*b*x^3 - 2*i)*e^(-1/2*(pi*b*x^3*(sign(f) - 1) + pi*a*(sign(f) -
1))*i)/(2*pi*b^2*i*ln(abs(f))*sign(f) - 2*pi*b^2*i*ln(abs(f)) - pi^2*b^2*sign(f)
+ pi^2*b^2 - 2*b^2*ln(abs(f))^2))*e^(b*x^3*ln(abs(f)) + a*ln(abs(f)))/i