3.41 \(\int (e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right ) \left (c+d x^n\right )^q \, dx\)

Optimal. Leaf size=211 \[ \frac{A (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{e (m+1)}+\frac{B x^{n+1} (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{m+n+1} \]

[Out]

(A*(e*x)^(1 + m)*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m)/n, -p, -q, (1 + m
+ n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(1 + m)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c
)^q) + (B*x^(1 + n)*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m + n)/n,
-p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m + n)*(1 + (b*x^n)/
a)^p*(1 + (d*x^n)/c)^q)

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Rubi [A]  time = 0.638616, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{A (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{e (m+1)}+\frac{B x^{n+1} (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{m+n+1} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(a + b*x^n)^p*(A + B*x^n)*(c + d*x^n)^q,x]

[Out]

(A*(e*x)^(1 + m)*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m)/n, -p, -q, (1 + m
+ n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(1 + m)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c
)^q) + (B*x^(1 + n)*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m + n)/n,
-p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m + n)*(1 + (b*x^n)/
a)^p*(1 + (d*x^n)/c)^q)

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Rubi in Sympy [A]  time = 65.9554, size = 170, normalized size = 0.81 \[ \frac{A \left (e x\right )^{m + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (1 + \frac{d x^{n}}{c}\right )^{- q} \left (a + b x^{n}\right )^{p} \left (c + d x^{n}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{m + 1}{n},- p,- q,\frac{m + n + 1}{n},- \frac{b x^{n}}{a},- \frac{d x^{n}}{c} \right )}}{e \left (m + 1\right )} + \frac{B x^{- m} x^{m + n + 1} \left (e x\right )^{m} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (1 + \frac{d x^{n}}{c}\right )^{- q} \left (a + b x^{n}\right )^{p} \left (c + d x^{n}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{m + n + 1}{n},- p,- q,\frac{m + 2 n + 1}{n},- \frac{b x^{n}}{a},- \frac{d x^{n}}{c} \right )}}{m + n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(a+b*x**n)**p*(A+B*x**n)*(c+d*x**n)**q,x)

[Out]

A*(e*x)**(m + 1)*(1 + b*x**n/a)**(-p)*(1 + d*x**n/c)**(-q)*(a + b*x**n)**p*(c +
d*x**n)**q*appellf1((m + 1)/n, -p, -q, (m + n + 1)/n, -b*x**n/a, -d*x**n/c)/(e*(
m + 1)) + B*x**(-m)*x**(m + n + 1)*(e*x)**m*(1 + b*x**n/a)**(-p)*(1 + d*x**n/c)*
*(-q)*(a + b*x**n)**p*(c + d*x**n)**q*appellf1((m + n + 1)/n, -p, -q, (m + 2*n +
1)/n, -b*x**n/a, -d*x**n/c)/(m + n + 1)

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Mathematica [B]  time = 1.93413, size = 458, normalized size = 2.17 \[ \frac{a c x (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (\frac{A (m+n+1)^2 F_1\left (\frac{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{(m+1) \left (n x^n \left (b c p F_1\left (\frac{m+n+1}{n};1-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a d q F_1\left (\frac{m+n+1}{n};-p,1-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )+a c (m+n+1) F_1\left (\frac{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )}+\frac{B (m+2 n+1) x^n F_1\left (\frac{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{n x^n \left (b c p F_1\left (\frac{m+2 n+1}{n};1-p,-q;\frac{m+3 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a d q F_1\left (\frac{m+2 n+1}{n};-p,1-q;\frac{m+3 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )+a c (m+2 n+1) F_1\left (\frac{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}\right )}{m+n+1} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^m*(a + b*x^n)^p*(A + B*x^n)*(c + d*x^n)^q,x]

[Out]

(a*c*x*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*((A*(1 + m + n)^2*AppellF1[(1 + m)/n,
-p, -q, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m)*(a*c*(1 + m + n)*A
ppellF1[(1 + m)/n, -p, -q, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)] + n*x^n*(b
*c*p*AppellF1[(1 + m + n)/n, 1 - p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)
/c)] + a*d*q*AppellF1[(1 + m + n)/n, -p, 1 - q, (1 + m + 2*n)/n, -((b*x^n)/a), -
((d*x^n)/c)]))) + (B*(1 + m + 2*n)*x^n*AppellF1[(1 + m + n)/n, -p, -q, (1 + m +
2*n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(a*c*(1 + m + 2*n)*AppellF1[(1 + m + n)/n,
-p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)] + n*x^n*(b*c*p*AppellF1[(1
+ m + 2*n)/n, 1 - p, -q, (1 + m + 3*n)/n, -((b*x^n)/a), -((d*x^n)/c)] + a*d*q*Ap
pellF1[(1 + m + 2*n)/n, -p, 1 - q, (1 + m + 3*n)/n, -((b*x^n)/a), -((d*x^n)/c)])
)))/(1 + m + n)

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Maple [F]  time = 0.665, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{q}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(a+b*x^n)^p*(A+B*x^n)*(c+d*x^n)^q,x)

[Out]

int((e*x)^m*(a+b*x^n)^p*(A+B*x^n)*(c+d*x^n)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m,x, algorithm="maxima")

[Out]

integrate((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m,x, algorithm="fricas")

[Out]

integral((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(a+b*x**n)**p*(A+B*x**n)*(c+d*x**n)**q,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m,x, algorithm="giac")

[Out]

integrate((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m, x)