3.354 \(\int F^{a+\frac{b}{(c+d x)^3}} (c+d x) \, dx\)

Optimal. Leaf size=49 \[ \frac{F^a (c+d x)^2 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[Out]

(F^a*(c + d*x)^2*Gamma[-2/3, -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^
3))^(2/3))/(3*d)

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Rubi [A]  time = 0.0462715, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{F^a (c+d x)^2 \left (-\frac{b \log (F)}{(c+d x)^3}\right )^{2/3} \text{Gamma}\left (-\frac{2}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(c + d*x)^2*Gamma[-2/3, -((b*Log[F])/(c + d*x)^3)]*(-((b*Log[F])/(c + d*x)^
3))^(2/3))/(3*d)

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Rubi in Sympy [A]  time = 4.26453, size = 48, normalized size = 0.98 \[ \frac{F^{a} \left (- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{3}}\right )^{\frac{2}{3}} \left (c + d x\right )^{2} \Gamma{\left (- \frac{2}{3},- \frac{b \log{\left (F \right )}}{\left (c + d x\right )^{3}} \right )}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

F**a*(-b*log(F)/(c + d*x)**3)**(2/3)*(c + d*x)**2*Gamma(-2/3, -b*log(F)/(c + d*x
)**3)/(3*d)

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Mathematica [A]  time = 0.0845763, size = 73, normalized size = 1.49 \[ \frac{F^a \left (\frac{b \log (F) \text{Gamma}\left (\frac{1}{3},-\frac{b \log (F)}{(c+d x)^3}\right )}{\sqrt [3]{-\frac{b \log (F)}{(c+d x)^3}}}+(c+d x)^3 F^{\frac{b}{(c+d x)^3}}\right )}{2 d (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

(F^a*(F^(b/(c + d*x)^3)*(c + d*x)^3 + (b*Gamma[1/3, -((b*Log[F])/(c + d*x)^3)]*L
og[F])/(-((b*Log[F])/(c + d*x)^3))^(1/3)))/(2*d*(c + d*x))

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \[ \int{F}^{a+{\frac{b}{ \left ( dx+c \right ) ^{3}}}} \left ( dx+c \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1}{2} \,{\left (F^{a} d x^{2} + 2 \, F^{a} c x\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} + \int \frac{3 \,{\left (F^{a} b d^{2} x^{2} \log \left (F\right ) + 2 \, F^{a} b c d x \log \left (F\right )\right )} F^{\frac{b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{2 \,{\left (d^{4} x^{4} + 4 \, c d^{3} x^{3} + 6 \, c^{2} d^{2} x^{2} + 4 \, c^{3} d x + c^{4}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(F^a*d*x^2 + 2*F^a*c*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)) + in
tegrate(3/2*(F^a*b*d^2*x^2*log(F) + 2*F^a*b*c*d*x*log(F))*F^(b/(d^3*x^3 + 3*c*d^
2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c
^4), x)

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Fricas [A]  time = 0.260288, size = 209, normalized size = 4.27 \[ \frac{F^{a} b \Gamma \left (\frac{1}{3}, -\frac{b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right ) +{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )} F^{\frac{a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}} \left (-\frac{b \log \left (F\right )}{d^{3}}\right )^{\frac{1}{3}}}{2 \, d^{2} \left (-\frac{b \log \left (F\right )}{d^{3}}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(F**(a+b/(d*x+c)**3)*(d*x+c),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (d x + c\right )} F^{a + \frac{b}{{\left (d x + c\right )}^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((d*x + c)*F^(a + b/(d*x + c)^3), x)