3.23 \(\int \frac{(e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right )}{c+d x^n} \, dx\)

Optimal. Leaf size=187 \[ \frac{(e x)^{m+1} \left (a^2 B d^2-2 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3 e (m+1)}-\frac{(e x)^{m+1} (b c-a d)^2 (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d^3 e (m+1)}-\frac{b x^{n+1} (e x)^m (-2 a B d-A b d+b B c)}{d^2 (m+n+1)}+\frac{b^2 B x^{2 n+1} (e x)^m}{d (m+2 n+1)} \]

[Out]

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

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Rubi [A]  time = 0.607289, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{(e x)^{m+1} \left (a^2 B d^2-2 a b d (B c-A d)+b^2 c (B c-A d)\right )}{d^3 e (m+1)}-\frac{(e x)^{m+1} (b c-a d)^2 (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d^3 e (m+1)}-\frac{b x^{n+1} (e x)^m (-2 a B d-A b d+b B c)}{d^2 (m+n+1)}+\frac{b^2 B x^{2 n+1} (e x)^m}{d (m+2 n+1)} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^m*(a + b*x^n)^2*(A + B*x^n))/(c + d*x^n),x]

[Out]

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

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Rubi in Sympy [A]  time = 66.2982, size = 214, normalized size = 1.14 \[ \frac{A a^{2} \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c e \left (m + 1\right )} + \frac{B b^{2} x^{- m} x^{m + 3 n + 1} \left (e x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 3 n + 1}{n} \\ \frac{m + 4 n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c \left (m + 3 n + 1\right )} + \frac{a x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (2 A b + B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c e \left (m + n + 1\right )} + \frac{b x^{2 n} \left (e x\right )^{- 2 n} \left (e x\right )^{m + 2 n + 1} \left (A b + 2 B a\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 2 n + 1}{n} \\ \frac{m + 3 n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c e \left (m + 2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(a+b*x**n)**2*(A+B*x**n)/(c+d*x**n),x)

[Out]

A*a**2*(e*x)**(m + 1)*hyper((1, (m + 1)/n), ((m + n + 1)/n,), -d*x**n/c)/(c*e*(m
+ 1)) + B*b**2*x**(-m)*x**(m + 3*n + 1)*(e*x)**m*hyper((1, (m + 3*n + 1)/n), ((
m + 4*n + 1)/n,), -d*x**n/c)/(c*(m + 3*n + 1)) + a*x**n*(e*x)**(-n)*(e*x)**(m +
n + 1)*(2*A*b + B*a)*hyper((1, (m + n + 1)/n), ((m + 2*n + 1)/n,), -d*x**n/c)/(c
*e*(m + n + 1)) + b*x**(2*n)*(e*x)**(-2*n)*(e*x)**(m + 2*n + 1)*(A*b + 2*B*a)*hy
per((1, (m + 2*n + 1)/n), ((m + 3*n + 1)/n,), -d*x**n/c)/(c*e*(m + 2*n + 1))

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Mathematica [A]  time = 0.371138, size = 156, normalized size = 0.83 \[ x (e x)^m \left (\frac{a^2 A}{c m+c}+\frac{(b c-a d)^2 (A d-B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c d^3 (m+1)}+\frac{(b c-a d)^2 (B c-A d)}{c d^3 (m+1)}+\frac{b x^n (2 a B d+A b d-b B c)}{d^2 (m+n+1)}+\frac{b^2 B x^{2 n}}{d m+2 d n+d}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^m*(a + b*x^n)^2*(A + B*x^n))/(c + d*x^n),x]

[Out]

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

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{2} \left ( A+B{x}^{n} \right ) }{c+d{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(a+b*x^n)^2*(A+B*x^n)/(c+d*x^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left ({\left (b^{2} c^{2} d e^{m} - 2 \, a b c d^{2} e^{m} + a^{2} d^{3} e^{m}\right )} A -{\left (b^{2} c^{3} e^{m} - 2 \, a b c^{2} d e^{m} + a^{2} c d^{2} e^{m}\right )} B\right )} \int \frac{x^{m}}{d^{4} x^{n} + c d^{3}}\,{d x} + \frac{{\left (m^{2} + m{\left (n + 2\right )} + n + 1\right )} B b^{2} d^{2} e^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )} -{\left ({\left ({\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} b^{2} c d e^{m} - 2 \,{\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a b d^{2} e^{m}\right )} A -{\left ({\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} b^{2} c^{2} e^{m} - 2 \,{\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a b c d e^{m} +{\left (m^{2} + m{\left (3 \, n + 2\right )} + 2 \, n^{2} + 3 \, n + 1\right )} a^{2} d^{2} e^{m}\right )} B\right )} x x^{m} +{\left ({\left (m^{2} + 2 \, m{\left (n + 1\right )} + 2 \, n + 1\right )} A b^{2} d^{2} e^{m} -{\left ({\left (m^{2} + 2 \, m{\left (n + 1\right )} + 2 \, n + 1\right )} b^{2} c d e^{m} - 2 \,{\left (m^{2} + 2 \, m{\left (n + 1\right )} + 2 \, n + 1\right )} a b d^{2} e^{m}\right )} B\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{{\left (m^{3} + 3 \, m^{2}{\left (n + 1\right )} +{\left (2 \, n^{2} + 6 \, n + 3\right )} m + 2 \, n^{2} + 3 \, n + 1\right )} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)^2*(e*x)^m/(d*x^n + c),x, algorithm="maxima")

[Out]

((b^2*c^2*d*e^m - 2*a*b*c*d^2*e^m + a^2*d^3*e^m)*A - (b^2*c^3*e^m - 2*a*b*c^2*d*
e^m + a^2*c*d^2*e^m)*B)*integrate(x^m/(d^4*x^n + c*d^3), x) + ((m^2 + m*(n + 2)
+ n + 1)*B*b^2*d^2*e^m*x*e^(m*log(x) + 2*n*log(x)) - (((m^2 + m*(3*n + 2) + 2*n^
2 + 3*n + 1)*b^2*c*d*e^m - 2*(m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*a*b*d^2*e^m)*
A - ((m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*b^2*c^2*e^m - 2*(m^2 + m*(3*n + 2) +
2*n^2 + 3*n + 1)*a*b*c*d*e^m + (m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*a^2*d^2*e^m
)*B)*x*x^m + ((m^2 + 2*m*(n + 1) + 2*n + 1)*A*b^2*d^2*e^m - ((m^2 + 2*m*(n + 1)
+ 2*n + 1)*b^2*c*d*e^m - 2*(m^2 + 2*m*(n + 1) + 2*n + 1)*a*b*d^2*e^m)*B)*x*e^(m*
log(x) + n*log(x)))/((m^3 + 3*m^2*(n + 1) + (2*n^2 + 6*n + 3)*m + 2*n^2 + 3*n +
1)*d^3)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B b^{2} x^{3 \, n} + A a^{2} +{\left (2 \, B a b + A b^{2}\right )} x^{2 \, n} +{\left (B a^{2} + 2 \, A a b\right )} x^{n}\right )} \left (e x\right )^{m}}{d x^{n} + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)^2*(e*x)^m/(d*x^n + c),x, algorithm="fricas")

[Out]

integral((B*b^2*x^(3*n) + A*a^2 + (2*B*a*b + A*b^2)*x^(2*n) + (B*a^2 + 2*A*a*b)*
x^n)*(e*x)^m/(d*x^n + c), 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)**2*(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. \[ \int \frac{{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{2} \left (e x\right )^{m}}{d x^{n} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(b*x^n + a)^2*(e*x)^m/(d*x^n + c),x, algorithm="giac")

[Out]

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