3.25 \(\int \frac {(e x)^m (A+B x^n)}{c+d x^n} \, dx\)

Optimal. Leaf size=78 \[ \frac {B (e x)^{m+1}}{d e (m+1)}-\frac {(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{c d e (m+1)} \]

[Out]

B*(e*x)^(1+m)/d/e/(1+m)-(-A*d+B*c)*(e*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-d*x^n/c)/c/d/e/(1+m)

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Rubi [A]  time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {459, 364} \[ \frac {B (e x)^{m+1}}{d e (m+1)}-\frac {(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )}{c d e (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(b*e*(m + n*(p + 1) + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]

Rubi steps

\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right )}{c+d x^n} \, dx &=\frac {B (e x)^{1+m}}{d e (1+m)}-\frac {(B c (1+m)-A d (1+m)) \int \frac {(e x)^m}{c+d x^n} \, dx}{d (1+m)}\\ &=\frac {B (e x)^{1+m}}{d e (1+m)}-\frac {(B c-A d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )}{c d e (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 57, normalized size = 0.73 \[ \frac {x (e x)^m \left ((A d-B c) \, _2F_1\left (1,\frac {m+1}{n};\frac {m+n+1}{n};-\frac {d x^n}{c}\right )+B c\right )}{c d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d x^{n} + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{d x^{n} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (e x \right )^{m}}{d \,x^{n}+c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {B e^{m} x x^{m}}{d {\left (m + 1\right )}} - {\left (B c e^{m} - A d e^{m}\right )} \int \frac {x^{m}}{d^{2} x^{n} + c d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )}{c+d\,x^n} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 4.09, size = 284, normalized size = 3.64 \[ \frac {A e^{m} m x x^{m} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{c n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {A e^{m} x x^{m} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{c n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {B e^{m} m x x^{m} x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{c n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B e^{m} x x^{m} x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{c n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B e^{m} x x^{m} x^{n} \Phi \left (\frac {d x^{n} e^{i \pi }}{c}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{c n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*e**m*m*x*x**m*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**2*gamma(m/n + 1 + 1/n)
) + A*e**m*x*x**m*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1/n)*gamma(m/n + 1/n)/(c*n**2*gamma(m/n + 1 + 1/
n)) + B*e**m*m*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(c*n**2*g
amma(m/n + 2 + 1/n)) + B*e**m*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gamma(m/n + 1 +
1/n)/(c*n*gamma(m/n + 2 + 1/n)) + B*e**m*x*x**m*x**n*lerchphi(d*x**n*exp_polar(I*pi)/c, 1, m/n + 1 + 1/n)*gam
ma(m/n + 1 + 1/n)/(c*n**2*gamma(m/n + 2 + 1/n))

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