3.1.5 \(\int \frac {(e x)^m (A+B x^n) (c+d x^n)}{a+b x^n} \, dx\) [5]

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

[Out]

B*d*x^(1+n)*(e*x)^m/b/(1+m+n)+(A*b*d-B*a*d+B*b*c)*(e*x)^(1+m)/b^2/e/(1+m)+(A*b-B*a)*(-a*d+b*c)*(e*x)^(1+m)*hyp
ergeom([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 = 0.08, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {584, 20, 30, 371} \begin {gather*} \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)} \end {gather*}

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))

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 584

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin {align*} \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )}{a+b x^n} \, dx &=\int \left (\frac {(b B c+A b d-a B d) (e x)^m}{b^2}+\frac {B d x^n (e x)^m}{b}+\frac {(A b-a B) (b c-a d) (e x)^m}{b^2 \left (a+b x^n\right )}\right ) \, dx\\ &=\frac {(b B c+A b d-a B d) (e x)^{1+m}}{b^2 e (1+m)}+\frac {(B d) \int x^n (e x)^m \, dx}{b}+\frac {((A b-a B) (b c-a d)) \int \frac {(e x)^m}{a+b x^n} \, dx}{b^2}\\ &=\frac {(b B c+A b d-a B d) (e x)^{1+m}}{b^2 e (1+m)}+\frac {(A b-a B) (b c-a d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^2 e (1+m)}+\frac {\left (B d x^{-m} (e x)^m\right ) \int x^{m+n} \, dx}{b}\\ &=\frac {B d x^{1+n} (e x)^m}{b (1+m+n)}+\frac {(b B c+A b d-a B d) (e x)^{1+m}}{b^2 e (1+m)}+\frac {(A b-a B) (b c-a d) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{a b^2 e (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 115, normalized size = 0.96 \begin {gather*} \frac {x (e x)^m \left (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) \, _2F_1\left (1,\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )\right )}{a b^2 (1+m) (1+m+n)} \end {gather*}

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 = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )}{a +b \,x^{n}}\, dx\]

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 = 0.00, 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)/(a+b*x^n),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) + ((m*e^m + e^m)*B
*b*d*x*e^(m*log(x) + n*log(x)) + ((m*e^m + (n + 1)*e^m)*A*b*d + ((m*e^m + (n + 1)*e^m)*b*c - (m*e^m + (n + 1)*
e^m)*a*d)*B)*x*x^m)/((m^2 + m*(n + 2) + n + 1)*b^2)

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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)/(a+b*x^n),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 5.67, size = 666, normalized size = 5.55 \begin {gather*} \frac {A c e^{m} m x x^{m} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {A c e^{m} x x^{m} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {A d e^{m} m x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {A d e^{m} x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {A d e^{m} x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B c e^{m} m x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B c e^{m} x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B c e^{m} x x^{m} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 1 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B d e^{m} m x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {2 B d e^{m} x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} + \frac {B d e^{m} x x^{m} x^{2 n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {m}{n} + 2 + \frac {1}{n}\right ) \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )}{a n^{2} \Gamma \left (\frac {m}{n} + 3 + \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)/(a+b*x**n),x)

[Out]

A*c*e**m*m*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*n**2*gamma(m/n + 1 + 1/
n)) + A*c*e**m*x*x**m*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(a*n**2*gamma(m/n + 1
+ 1/n)) + A*d*e**m*m*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*
n**2*gamma(m/n + 2 + 1/n)) + A*d*e**m*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m
/n + 1 + 1/n)/(a*n*gamma(m/n + 2 + 1/n)) + A*d*e**m*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1
+ 1/n)*gamma(m/n + 1 + 1/n)/(a*n**2*gamma(m/n + 2 + 1/n)) + B*c*e**m*m*x*x**m*x**n*lerchphi(b*x**n*exp_polar(I
*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*n**2*gamma(m/n + 2 + 1/n)) + B*c*e**m*x*x**m*x**n*lerchphi(b
*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*n*gamma(m/n + 2 + 1/n)) + B*c*e**m*x*x**m*x
**n*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(a*n**2*gamma(m/n + 2 + 1/n)) +
B*d*e**m*m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*n**2*g
amma(m/n + 3 + 1/n)) + 2*B*d*e**m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m
/n + 2 + 1/n)/(a*n*gamma(m/n + 3 + 1/n)) + B*d*e**m*x*x**m*x**(2*n)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n
+ 2 + 1/n)*gamma(m/n + 2 + 1/n)/(a*n**2*gamma(m/n + 3 + 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)/(a+b*x^n),x, algorithm="giac")

[Out]

integrate((B*x^n + A)*(d*x^n + c)*(x*e)^m/(b*x^n + a), 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 )\,\left (c+d\,x^n\right )}{a+b\,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))/(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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