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

Optimal. Leaf size=271 \[ -\frac{(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) (A b (m+n p+n+1) (a d (m+1)-b c (m+n (p+2)+1))-a (m+1) (a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1))))}{b^2 e (m+1) (m+n p+n+1) (m+n (p+2)+1)}-\frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1} (a B d (m+n+1)-b (A d n+B c (m+n (p+2)+1)))}{b^2 e (m+n p+n+1) (m+n (p+2)+1)}+\frac{d (e x)^{m+1} \left (A+B x^n\right ) \left (a+b x^n\right )^{p+1}}{b e (m+n (p+2)+1)} \]

[Out]

-(((a*B*d*(1 + m + n) - b*(A*d*n + B*c*(1 + m + n*(2 + p))))*(e*x)^(1 + m)*(a +
b*x^n)^(1 + p))/(b^2*e*(1 + m + n + n*p)*(1 + m + n*(2 + p)))) + (d*(e*x)^(1 + m
)*(a + b*x^n)^(1 + p)*(A + B*x^n))/(b*e*(1 + m + n*(2 + p))) - ((A*b*(1 + m + n
+ n*p)*(a*d*(1 + m) - b*c*(1 + m + n*(2 + p))) - a*(1 + m)*(a*B*d*(1 + m + n) -
b*(A*d*n + B*c*(1 + m + n*(2 + p)))))*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric
2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(b^2*e*(1 + m)*(1 + m + n + n*p
)*(1 + m + n*(2 + p))*(1 + (b*x^n)/a)^p)

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Rubi [A]  time = 0.806936, antiderivative size = 255, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1} (-a B d (m+n+1)+A b d n+b B c (m+n (p+2)+1))}{b^2 e (m+n p+n+1) (m+n (p+2)+1)}-\frac{(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right ) \left (\frac{a (-a B d (m+n+1)+A b d n+b B c (m+n (p+2)+1))}{b (m+n p+n+1)}+a A d-\frac{A b c (m+n (p+2)+1)}{m+1}\right )}{b e (m+n (p+2)+1)}+\frac{d (e x)^{m+1} \left (A+B x^n\right ) \left (a+b x^n\right )^{p+1}}{b e (m+n (p+2)+1)} \]

Antiderivative was successfully verified.

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

[Out]

((A*b*d*n - a*B*d*(1 + m + n) + b*B*c*(1 + m + n*(2 + p)))*(e*x)^(1 + m)*(a + b*
x^n)^(1 + p))/(b^2*e*(1 + m + n + n*p)*(1 + m + n*(2 + p))) + (d*(e*x)^(1 + m)*(
a + b*x^n)^(1 + p)*(A + B*x^n))/(b*e*(1 + m + n*(2 + p))) - ((a*A*d - (A*b*c*(1
+ m + n*(2 + p)))/(1 + m) + (a*(A*b*d*n - a*B*d*(1 + m + n) + b*B*c*(1 + m + n*(
2 + p))))/(b*(1 + m + n + n*p)))*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(
1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(b*e*(1 + m + n*(2 + p))*(1 + (b*x^n
)/a)^p)

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Rubi in Sympy [A]  time = 119.87, size = 246, normalized size = 0.91 \[ \frac{d \left (e x\right )^{m + 1} \left (A + B x^{n}\right ) \left (a + b x^{n}\right )^{p + 1}}{b e \left (m + n \left (p + 2\right ) + 1\right )} - \frac{\left (e x\right )^{m + 1} \left (a + b x^{n}\right )^{p + 1} \left (- B b c n \left (p + 2\right ) + B \left (m + 1\right ) \left (a d - b c\right ) - d n \left (A b - B a\right )\right )}{b^{2} e \left (m + n \left (p + 1\right ) + 1\right ) \left (m + n \left (p + 2\right ) + 1\right )} + \frac{\left (e x\right )^{m + 1} \left (1 + \frac{b x^{n}}{a}\right )^{- p} \left (a + b x^{n}\right )^{p} \left (- A b \left (- b c n \left (p + 2\right ) + \left (m + 1\right ) \left (a d - b c\right )\right ) \left (m + n \left (p + 1\right ) + 1\right ) + a \left (m + 1\right ) \left (- B b c n \left (p + 2\right ) + B \left (m + 1\right ) \left (a d - b c\right ) - d n \left (A b - B a\right )\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{b^{2} e \left (m + 1\right ) \left (m + n \left (p + 1\right ) + 1\right ) \left (m + n \left (p + 2\right ) + 1\right )} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.948179, size = 164, normalized size = 0.61 \[ x (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (x^n \left (\frac{(A d+B c) \, _2F_1\left (\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+\frac{B d x^n \, _2F_1\left (\frac{m+2 n+1}{n},-p;\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{m+2 n+1}\right )+\frac{A c \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*(a + b*x^n)^p*((A*c*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -
((b*x^n)/a)])/(1 + m) + x^n*(((B*c + A*d)*Hypergeometric2F1[(1 + m + n)/n, -p, (
1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n) + (B*d*x^n*Hypergeometric2F1[(1 + m +
2*n)/n, -p, (1 + m + 3*n)/n, -((b*x^n)/a)])/(1 + m + 2*n))))/(1 + (b*x^n)/a)^p

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Maple [F]  time = 0.11, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) \, dx \]

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),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (B d x^{2 \, n} + A c +{\left (B c + A d\right )} x^{n}\right )}{\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*d*x^(2*n) + A*c + (B*c + A*d)*x^n)*(b*x^n + a)^p*(e*x)^m, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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