3.296 \(\int \frac{F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx\)

Optimal. Leaf size=49 \[ -\frac{F^a \sqrt [3]{-b \log (F) (c+d x)^3} \text{Gamma}\left (-\frac{1}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.0643983, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2218} \[ -\frac{F^a \sqrt [3]{-b \log (F) (c+d x)^3} \text{Gamma}\left (-\frac{1}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{F^{a+b (c+d x)^3}}{(c+d x)^2} \, dx &=-\frac{F^a \Gamma \left (-\frac{1}{3},-b (c+d x)^3 \log (F)\right ) \sqrt [3]{-b (c+d x)^3 \log (F)}}{3 d (c+d x)}\\ \end{align*}

Mathematica [A]  time = 0.0159363, size = 49, normalized size = 1. \[ -\frac{F^a \sqrt [3]{-b \log (F) (c+d x)^3} \text{Gamma}\left (-\frac{1}{3},-b \log (F) (c+d x)^3\right )}{3 d (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{F}^{a+b \left ( dx+c \right ) ^{3}}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x)

[Out]

int(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^2, x)

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Fricas [B]  time = 1.54923, size = 250, normalized size = 5.1 \begin{align*} \frac{\left (-b d^{3} \log \left (F\right )\right )^{\frac{1}{3}}{\left (d x + c\right )} F^{a} \Gamma \left (\frac{2}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) - F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a} d}{d^{3} x + c d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**2,x)

[Out]

Integral(F**(a + b*(c + d*x)**3)/(c + d*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^2, x)