3.300 \(\int f^{a+b \sqrt [3]{c+d x}} \, dx\)

Optimal. Leaf size=100 \[ \frac{6 f^{a+b \sqrt [3]{c+d x}}}{b^3 d \log ^3(f)}-\frac{6 \sqrt [3]{c+d x} f^{a+b \sqrt [3]{c+d x}}}{b^2 d \log ^2(f)}+\frac{3 (c+d x)^{2/3} f^{a+b \sqrt [3]{c+d x}}}{b d \log (f)} \]

[Out]

(6*f^(a + b*(c + d*x)^(1/3)))/(b^3*d*Log[f]^3) - (6*f^(a + b*(c + d*x)^(1/3))*(c
+ d*x)^(1/3))/(b^2*d*Log[f]^2) + (3*f^(a + b*(c + d*x)^(1/3))*(c + d*x)^(2/3))/
(b*d*Log[f])

_______________________________________________________________________________________

Rubi [A]  time = 0.115282, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{6 f^{a+b \sqrt [3]{c+d x}}}{b^3 d \log ^3(f)}-\frac{6 \sqrt [3]{c+d x} f^{a+b \sqrt [3]{c+d x}}}{b^2 d \log ^2(f)}+\frac{3 (c+d x)^{2/3} f^{a+b \sqrt [3]{c+d x}}}{b d \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*(c + d*x)^(1/3)),x]

[Out]

(6*f^(a + b*(c + d*x)^(1/3)))/(b^3*d*Log[f]^3) - (6*f^(a + b*(c + d*x)^(1/3))*(c
+ d*x)^(1/3))/(b^2*d*Log[f]^2) + (3*f^(a + b*(c + d*x)^(1/3))*(c + d*x)^(2/3))/
(b*d*Log[f])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.247, size = 88, normalized size = 0.88 \[ \frac{3 f^{a + b \sqrt [3]{c + d x}} \left (c + d x\right )^{\frac{2}{3}}}{b d \log{\left (f \right )}} - \frac{6 f^{a + b \sqrt [3]{c + d x}} \sqrt [3]{c + d x}}{b^{2} d \log{\left (f \right )}^{2}} + \frac{6 f^{a + b \sqrt [3]{c + d x}}}{b^{3} d \log{\left (f \right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*f**(a + b*(c + d*x)**(1/3))*(c + d*x)**(2/3)/(b*d*log(f)) - 6*f**(a + b*(c + d
*x)**(1/3))*(c + d*x)**(1/3)/(b**2*d*log(f)**2) + 6*f**(a + b*(c + d*x)**(1/3))/
(b**3*d*log(f)**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0441458, size = 60, normalized size = 0.6 \[ \frac{3 f^{a+b \sqrt [3]{c+d x}} \left (b^2 \log ^2(f) (c+d x)^{2/3}-2 b \log (f) \sqrt [3]{c+d x}+2\right )}{b^3 d \log ^3(f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*(c + d*x)^(1/3)),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.005, size = 0, normalized size = 0. \[ \int{f}^{a+b\sqrt [3]{dx+c}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b*(d*x+c)^(1/3)),x)

[Out]

int(f^(a+b*(d*x+c)^(1/3)),x)

_______________________________________________________________________________________

Maxima [A]  time = 0.952551, size = 84, normalized size = 0.84 \[ \frac{3 \,{\left ({\left (d x + c\right )}^{\frac{2}{3}} b^{2} f^{a} \log \left (f\right )^{2} - 2 \,{\left (d x + c\right )}^{\frac{1}{3}} b f^{a} \log \left (f\right ) + 2 \, f^{a}\right )} f^{{\left (d x + c\right )}^{\frac{1}{3}} b}}{b^{3} d \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^((d*x + c)^(1/3)*b + a),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.252559, size = 78, normalized size = 0.78 \[ \frac{3 \,{\left ({\left (d x + c\right )}^{\frac{2}{3}} b^{2} \log \left (f\right )^{2} - 2 \,{\left (d x + c\right )}^{\frac{1}{3}} b \log \left (f\right ) + 2\right )} e^{\left ({\left (d x + c\right )}^{\frac{1}{3}} b \log \left (f\right ) + a \log \left (f\right )\right )}}{b^{3} d \log \left (f\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^((d*x + c)^(1/3)*b + a),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{a + b \sqrt [3]{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(f**(a + b*(c + d*x)**(1/3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.305435, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^((d*x + c)^(1/3)*b + a),x, algorithm="giac")

[Out]

Done