3.3.94 \(\int F^{a+b (c+d x)^3} (c+d x) \, dx\) [294]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2250} \begin {gather*} -\frac {F^a (c+d x)^2 \text {Gamma}\left (\frac {2}{3},-b \log (F) (c+d x)^3\right )}{3 d \left (-b \log (F) (c+d x)^3\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2250

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

Rubi steps

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

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Mathematica [A]
time = 0.16, size = 49, normalized size = 1.00 \begin {gather*} -\frac {F^a (c+d x)^2 \Gamma \left (\frac {2}{3},-b (c+d x)^3 \log (F)\right )}{3 d \left (-b (c+d x)^3 \log (F)\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int F^{a +b \left (d x +c \right )^{3}} \left (d x +c \right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.10, size = 63, normalized size = 1.29 \begin {gather*} \frac {\left (-b d^{3} \log \left (F\right )\right )^{\frac {1}{3}} 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 )}{3 \, b d^{2} \log \left (F\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-b*d^3*log(F))^(1/3)*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))/(b*d^2*log
(F))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int F^{a + b \left (c + d x\right )^{3}} \left (c + d x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int F^{a+b\,{\left (c+d\,x\right )}^3}\,\left (c+d\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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