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

Optimal. Leaf size=120 \[ \frac{(e x)^{m+1} (A b-a B) (b c-a d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a b^2 e (m+1)}+\frac{(e x)^{m+1} (-a B d+A b d+b B c)}{b^2 e (m+1)}+\frac{B d x^{n+1} (e x)^m}{b (m+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.32854, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{(e x)^{m+1} (A b-a B) (b c-a d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a b^2 e (m+1)}+\frac{(e x)^{m+1} (-a B d+A b d+b B c)}{b^2 e (m+1)}+\frac{B d x^{n+1} (e x)^m}{b (m+n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 40.0225, size = 141, normalized size = 1.18 \[ \frac{A c \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + 1\right )} + \frac{B d x^{- m} x^{m + 2 n + 1} \left (e x\right )^{m}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 2 n + 1}{n} \\ \frac{m + 3 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \left (m + 2 n + 1\right )} + \frac{x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (A d + B c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.183099, size = 115, normalized size = 0.96 \[ \frac{x (e x)^m \left ((m+n+1) (A b-a B) (b c-a d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )-(m+n+1) (A b-a B) (b c-a d)+a b B d (m+1) x^n+A b^2 c (m+n+1)\right )}{a b^2 (m+1) (m+n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }{a+b{x}^{n}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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