3.277 \(\int (e x)^{7/4} (c+d x^n)^q (a x^j+b x^{j+n})^{5/3} \, dx\)

Optimal. Leaf size=129 \[ \frac{12 a e (e x)^{3/4} x^{j+2} \left (a x^j+b x^{j+n}\right )^{2/3} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{20 j+33}{12 n};-\frac{5}{3},-q;\frac{20 j+12 n+33}{12 n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{(20 j+33) \left (\frac{b x^n}{a}+1\right )^{2/3}} \]

[Out]

(12*a*e*x^(2 + j)*(e*x)^(3/4)*(c + d*x^n)^q*(a*x^j + b*x^(j + n))^(2/3)*AppellF1[(33 + 20*j)/(12*n), -5/3, -q,
(33 + 20*j + 12*n)/(12*n), -((b*x^n)/a), -((d*x^n)/c)])/((33 + 20*j)*(1 + (b*x^n)/a)^(2/3)*(1 + (d*x^n)/c)^q)

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Rubi [A]  time = 0.356796, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.088, Rules used = {2042, 511, 510} \[ \frac{12 a e (e x)^{3/4} x^{j+2} \left (a x^j+b x^{j+n}\right )^{2/3} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{20 j+33}{12 n};-\frac{5}{3},-q;\frac{20 j+12 n+33}{12 n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{(20 j+33) \left (\frac{b x^n}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(7/4)*(c + d*x^n)^q*(a*x^j + b*x^(j + n))^(5/3),x]

[Out]

(12*a*e*x^(2 + j)*(e*x)^(3/4)*(c + d*x^n)^q*(a*x^j + b*x^(j + n))^(2/3)*AppellF1[(33 + 20*j)/(12*n), -5/3, -q,
(33 + 20*j + 12*n)/(12*n), -((b*x^n)/a), -((d*x^n)/c)])/((33 + 20*j)*(1 + (b*x^n)/a)^(2/3)*(1 + (d*x^n)/c)^q)

Rule 2042

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol]
:> Dist[(e^IntPart[m]*(e*x)^FracPart[m]*(a*x^j + b*x^(j + n))^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a
+ b*x^n)^FracPart[p]), Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n,
p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int (e x)^{7/4} \left (c+d x^n\right )^q \left (a x^j+b x^{j+n}\right )^{5/3} \, dx &=\frac{\left (e x^{-\frac{3}{4}-\frac{2 j}{3}} (e x)^{3/4} \left (a x^j+b x^{j+n}\right )^{2/3}\right ) \int x^{\frac{7}{4}+\frac{5 j}{3}} \left (a+b x^n\right )^{5/3} \left (c+d x^n\right )^q \, dx}{\left (a+b x^n\right )^{2/3}}\\ &=\frac{\left (a e x^{-\frac{3}{4}-\frac{2 j}{3}} (e x)^{3/4} \left (a x^j+b x^{j+n}\right )^{2/3}\right ) \int x^{\frac{7}{4}+\frac{5 j}{3}} \left (1+\frac{b x^n}{a}\right )^{5/3} \left (c+d x^n\right )^q \, dx}{\left (1+\frac{b x^n}{a}\right )^{2/3}}\\ &=\frac{\left (a e x^{-\frac{3}{4}-\frac{2 j}{3}} (e x)^{3/4} \left (c+d x^n\right )^q \left (1+\frac{d x^n}{c}\right )^{-q} \left (a x^j+b x^{j+n}\right )^{2/3}\right ) \int x^{\frac{7}{4}+\frac{5 j}{3}} \left (1+\frac{b x^n}{a}\right )^{5/3} \left (1+\frac{d x^n}{c}\right )^q \, dx}{\left (1+\frac{b x^n}{a}\right )^{2/3}}\\ &=\frac{12 a e x^{2+j} (e x)^{3/4} \left (c+d x^n\right )^q \left (1+\frac{d x^n}{c}\right )^{-q} \left (a x^j+b x^{j+n}\right )^{2/3} F_1\left (\frac{33+20 j}{12 n};-\frac{5}{3},-q;\frac{33+20 j+12 n}{12 n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{(33+20 j) \left (1+\frac{b x^n}{a}\right )^{2/3}}\\ \end{align*}

Mathematica [A]  time = 0.211557, size = 210, normalized size = 1.63 \[ \frac{12 (e x)^{7/4} x^{j+1} \left (x^j \left (a+b x^n\right )\right )^{2/3} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} \left (b (20 j+33) x^n F_1\left (\frac{20 j+12 n+33}{12 n};-\frac{2}{3},-q;\frac{20 j+24 n+33}{12 n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a (20 j+12 n+33) F_1\left (\frac{20 j+33}{12 n};-\frac{2}{3},-q;\frac{\frac{5 j}{3}+n+\frac{11}{4}}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )}{(20 j+33) (20 j+12 n+33) \left (\frac{b x^n}{a}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*x)^(7/4)*(c + d*x^n)^q*(a*x^j + b*x^(j + n))^(5/3),x]

[Out]

(12*x^(1 + j)*(e*x)^(7/4)*(x^j*(a + b*x^n))^(2/3)*(c + d*x^n)^q*(a*(33 + 20*j + 12*n)*AppellF1[(33 + 20*j)/(12
*n), -2/3, -q, (11/4 + (5*j)/3 + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + b*(33 + 20*j)*x^n*AppellF1[(33 + 20*j + 1
2*n)/(12*n), -2/3, -q, (33 + 20*j + 24*n)/(12*n), -((b*x^n)/a), -((d*x^n)/c)]))/((33 + 20*j)*(33 + 20*j + 12*n
)*(1 + (b*x^n)/a)^(2/3)*(1 + (d*x^n)/c)^q)

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Maple [F]  time = 0.823, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{{\frac{7}{4}}} \left ( c+d{x}^{n} \right ) ^{q} \left ( a{x}^{j}+b{x}^{j+n} \right ) ^{{\frac{5}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(7/4)*(c+d*x^n)^q*(a*x^j+b*x^(j+n))^(5/3),x)

[Out]

int((e*x)^(7/4)*(c+d*x^n)^q*(a*x^j+b*x^(j+n))^(5/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{j + n} + a x^{j}\right )}^{\frac{5}{3}} \left (e x\right )^{\frac{7}{4}}{\left (d x^{n} + c\right )}^{q}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/4)*(c+d*x^n)^q*(a*x^j+b*x^(j+n))^(5/3),x, algorithm="maxima")

[Out]

integrate((b*x^(j + n) + a*x^j)^(5/3)*(e*x)^(7/4)*(d*x^n + c)^q, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b e x x^{j + n} + a e x x^{j}\right )}{\left (b x^{j + n} + a x^{j}\right )}^{\frac{2}{3}} \left (e x\right )^{\frac{3}{4}}{\left (d x^{n} + c\right )}^{q}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/4)*(c+d*x^n)^q*(a*x^j+b*x^(j+n))^(5/3),x, algorithm="fricas")

[Out]

integral((b*e*x*x^(j + n) + a*e*x*x^j)*(b*x^(j + n) + a*x^j)^(2/3)*(e*x)^(3/4)*(d*x^n + c)^q, 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((e*x)**(7/4)*(c+d*x**n)**q*(a*x**j+b*x**(j+n))**(5/3),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{j + n} + a x^{j}\right )}^{\frac{5}{3}} \left (e x\right )^{\frac{7}{4}}{\left (d x^{n} + c\right )}^{q}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(7/4)*(c+d*x^n)^q*(a*x^j+b*x^(j+n))^(5/3),x, algorithm="giac")

[Out]

integrate((b*x^(j + n) + a*x^j)^(5/3)*(e*x)^(7/4)*(d*x^n + c)^q, x)