3.1.25 \(\int \frac {(e x)^m (A+B x^n)}{c+d x^n} \, dx\) [25]

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

[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 = 0.03, 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 = {470, 371} \begin {gather*} \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)} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 470

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 = 0.09, size = 59, normalized size = 0.76 \begin {gather*} -\frac {x (e x)^m \left (-B c+(B c-A d) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {d x^n}{c}\right )\right )}{c d (1+m)} \end {gather*}

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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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right )}{c +d \,x^{n}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)*(x*e)^m/(d*x^n + c), x)

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Sympy [C] Result contains complex when optimal does not.
time = 2.00, size = 284, normalized size = 3.64 \begin {gather*} \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 )} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)*(x*e)^m/(d*x^n + c), x)

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

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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