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]

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

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Rubi [A]  time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.07, 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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fricas [F]  time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b x} {\left (\pi + d x\right )}^{n} {\left (f x + E\right )}^{p} b^{2} x^{2}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b x\right )^{\frac {5}{2}} {\left (\pi + d x\right )}^{n} {\left (f x + E\right )}^{p}\,{d x} \]

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)

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maple [F]  time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (b x \right )^{\frac {5}{2}} \left (f x +E \right )^{p} \left (d x +\pi \right )^{n}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b x\right )^{\frac {5}{2}} {\left (\pi + d x\right )}^{n} {\left (f x + E\right )}^{p}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\mathrm {e}+f\,x\right )}^p\,{\left (b\,x\right )}^{5/2}\,{\left (\Pi +d\,x\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1) + f*x)^p*(b*x)^(5/2)*(Pi + d*x)^n,x)

[Out]

int((exp(1) + f*x)^p*(b*x)^(5/2)*(Pi + d*x)^n, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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