3.968 \(\int (b x)^{5/2} (\pi +d x)^n (e+f x)^p \, dx\)

Optimal. Leaf size=47 \[ \frac{2 \pi ^n e^p (b x)^{7/2} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b} \]

[Out]

(2*E^p*Pi^n*(b*x)^(7/2)*AppellF1[7/2, -n, -p, 9/2, -((d*x)/Pi), -((f*x)/E)])/(7*b)

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Rubi [A]  time = 0.0147463, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {133} \[ \frac{2 \pi ^n e^p (b x)^{7/2} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(b*x)^(5/2)*(Pi + d*x)^n*(E + f*x)^p,x]

[Out]

(2*E^p*Pi^n*(b*x)^(7/2)*AppellF1[7/2, -n, -p, 9/2, -((d*x)/Pi), -((f*x)/E)])/(7*b)

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int (b x)^{5/2} (\pi +d x)^n (e+f x)^p \, dx &=\frac{2 e^p \pi ^n (b x)^{7/2} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right )}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0691104, size = 45, normalized size = 0.96 \[ \frac{2}{7} \pi ^n e^p x (b x)^{5/2} F_1\left (\frac{7}{2};-n,-p;\frac{9}{2};-\frac{d x}{\pi },-\frac{f x}{e}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x)^(5/2)*(Pi + d*x)^n*(E + f*x)^p,x]

[Out]

(2*E^p*Pi^n*x*(b*x)^(5/2)*AppellF1[7/2, -n, -p, 9/2, -((d*x)/Pi), -((f*x)/E)])/7

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{{\frac{5}{2}}} \left ( dx+\pi \right ) ^{n} \left ( fx+E \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^(5/2)*(d*x+Pi)^n*(f*x+E)^p,x)

[Out]

int((b*x)^(5/2)*(d*x+Pi)^n*(f*x+E)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x\right )^{\frac{5}{2}}{\left (\pi + d x\right )}^{n}{\left (f x + E\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^(5/2)*(d*x+pi)^n*(f*x+E)^p,x, algorithm="maxima")

[Out]

integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + E)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b x}{\left (\pi + d x\right )}^{n}{\left (f x + E\right )}^{p} b^{2} x^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^(5/2)*(d*x+pi)^n*(f*x+E)^p,x, algorithm="fricas")

[Out]

integral(sqrt(b*x)*(pi + d*x)^n*(f*x + E)^p*b^2*x^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**(5/2)*(d*x+pi)**n*(f*x+E)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b x\right )^{\frac{5}{2}}{\left (\pi + d x\right )}^{n}{\left (f x + E\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^(5/2)*(d*x+pi)^n*(f*x+E)^p,x, algorithm="giac")

[Out]

integrate((b*x)^(5/2)*(pi + d*x)^n*(f*x + E)^p, x)