3.1.33 \(\int \frac {(e x)^m (A+B x^2)}{(c+d x^2)^2} \, dx\) [33]

Optimal. Leaf size=103 \[ -\frac {(B c-A d) (e x)^{1+m}}{2 c d e \left (c+d x^2\right )}+\frac {(A d (1-m)+B c (1+m)) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{2 c^2 d e (1+m)} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 103, 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 = {468, 371} \begin {gather*} \frac {(e x)^{m+1} (A d (1-m)+B c (m+1)) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {d x^2}{c}\right )}{2 c^2 d e (m+1)}-\frac {(e x)^{m+1} (B c-A d)}{2 c d e \left (c+d x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((B*c - A*d)*(e*x)^(1 + m))/(c*d*e*(c + d*x^2)) + ((A*d*(1 - m) + B*c*(1 + m))*(e*x)^(1 + m)*Hypergeometr
ic2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(2*c^2*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 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d
))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b*e*n*(p + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a
*b*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0]
&& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0]
&& LeQ[-1, m, (-n)*(p + 1)]))

Rubi steps

\begin {align*} \int \frac {(e x)^m \left (A+B x^2\right )}{\left (c+d x^2\right )^2} \, dx &=-\frac {(B c-A d) (e x)^{1+m}}{2 c d e \left (c+d x^2\right )}+\frac {(-A d (-1+m)+B c (1+m)) \int \frac {(e x)^m}{c+d x^2} \, dx}{2 c d}\\ &=-\frac {(B c-A d) (e x)^{1+m}}{2 c d e \left (c+d x^2\right )}+\frac {(A d (1-m)+B c (1+m)) (e x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )}{2 c^2 d e (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 81, normalized size = 0.79 \begin {gather*} \frac {x (e x)^m \left (B c \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )+(-B c+A d) \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {d x^2}{c}\right )\right )}{c^2 d (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{m} \left (B \,x^{2}+A \right )}{\left (d \,x^{2}+c \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x)^m*(B*x^2+A)/(d*x^2+c)^2,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*(B*x^2+A)/(d*x^2+c)^2,x, algorithm="maxima")

[Out]

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

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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*(B*x^2+A)/(d*x^2+c)^2,x, algorithm="fricas")

[Out]

integral((B*x^2 + A)*(x*e)^m/(d^2*x^4 + 2*c*d*x^2 + c^2), x)

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Sympy [C] Result contains complex when optimal does not.
time = 15.40, size = 954, normalized size = 9.26 \begin {gather*} A \left (- \frac {c e^{m} m^{2} x x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 c e^{m} m x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {c e^{m} x x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {2 c e^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {d e^{m} m^{2} x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d e^{m} x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {1}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {1}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}\right ) + B \left (- \frac {c e^{m} m^{2} x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {4 c e^{m} m x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {2 c e^{m} m x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {3 c e^{m} x^{3} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {6 c e^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {d e^{m} m^{2} x^{5} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {4 d e^{m} m x^{5} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {3 d e^{m} x^{5} x^{m} \Phi \left (\frac {d x^{2} e^{i \pi }}{c}, 1, \frac {m}{2} + \frac {3}{2}\right ) \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )}{8 c^{3} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 c^{2} d x^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(B*x**2+A)/(d*x**2+c)**2,x)

[Out]

A*(-c*e**m*m**2*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*c**3*gamma(m/2 + 3
/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2)) + 2*c*e**m*m*x*x**m*gamma(m/2 + 1/2)/(8*c**3*gamma(m/2 + 3/2) + 8*c**2*d
*x**2*gamma(m/2 + 3/2)) + c*e**m*x*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*c
**3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2)) + 2*c*e**m*x*x**m*gamma(m/2 + 1/2)/(8*c**3*gamma(m/2 +
3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2)) - d*e**m*m**2*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 1/2
)*gamma(m/2 + 1/2)/(8*c**3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2)) + d*e**m*x**3*x**m*lerchphi(d*x*
*2*exp_polar(I*pi)/c, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(8*c**3*gamma(m/2 + 3/2) + 8*c**2*d*x**2*gamma(m/2 + 3/2)
)) + B*(-c*e**m*m**2*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma
(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - 4*c*e**m*m*x**3*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2
+ 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) + 2*c*e**m*m*x**3*x**m*gam
ma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - 3*c*e**m*x**3*x**m*lerchphi(d*x**2*
exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) +
6*c*e**m*x**3*x**m*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - d*e**m*m**2*
x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*
d*x**2*gamma(m/2 + 5/2)) - 4*d*e**m*m*x**5*x**m*lerchphi(d*x**2*exp_polar(I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3
/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)) - 3*d*e**m*x**5*x**m*lerchphi(d*x**2*exp_polar(
I*pi)/c, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(8*c**3*gamma(m/2 + 5/2) + 8*c**2*d*x**2*gamma(m/2 + 5/2)))

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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*(B*x^2+A)/(d*x^2+c)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (e\,x\right )}^m}{{\left (d\,x^2+c\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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