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(1/12*(33+20*j)/n,-5/3,-q,1+(11/4+5/3*j
)/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)

________________________________________________________________________________________

Rubi [A]  time = 0.36, antiderivative size = 129, normalized size of antiderivative = 1.00, 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 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])

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 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])

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 = 0.25, 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)

________________________________________________________________________________________

fricas [F]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b e x x^{j + n} + a e x x^{j}\right )} {\left (b x^{j + n} + a x^{j}\right )}^{\frac {2}{3}} \left (e x\right )^{\frac {3}{4}} {\left (d x^{n} + c\right )}^{q}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{j + n} + a x^{j}\right )}^{\frac {5}{3}} \left (e x\right )^{\frac {7}{4}} {\left (d x^{n} + c\right )}^{q}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 1.08, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{\frac {7}{4}} \left (a \,x^{j}+b \,x^{j +n}\right )^{\frac {5}{3}} \left (d \,x^{n}+c \right )^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{j + n} + a x^{j}\right )}^{\frac {5}{3}} \left (e x\right )^{\frac {7}{4}} {\left (d x^{n} + c\right )}^{q}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,x^j+b\,x^{j+n}\right )}^{5/3}\,{\left (e\,x\right )}^{7/4}\,{\left (c+d\,x^n\right )}^q \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________