3.4.24 \(\int \frac {x^m (A+B x^2)}{(a+b x^2)^3} \, dx\) [324]

Optimal. Leaf size=93 \[ \frac {(A b-a B) x^{1+m}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b (3-m)+a B (1+m)) x^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{4 a^3 b (1+m)} \]

[Out]

1/4*(A*b-B*a)*x^(1+m)/a/b/(b*x^2+a)^2+1/4*(A*b*(3-m)+a*B*(1+m))*x^(1+m)*hypergeom([2, 1/2+1/2*m],[3/2+1/2*m],-
b*x^2/a)/a^3/b/(1+m)

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Rubi [A]
time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {468, 371} \begin {gather*} \frac {x^{m+1} (a B (m+1)+A b (3-m)) \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};-\frac {b x^2}{a}\right )}{4 a^3 b (m+1)}+\frac {x^{m+1} (A b-a B)}{4 a b \left (a+b x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^m*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

((A*b - a*B)*x^(1 + m))/(4*a*b*(a + b*x^2)^2) + ((A*b*(3 - m) + a*B*(1 + m))*x^(1 + m)*Hypergeometric2F1[2, (1
+ m)/2, (3 + m)/2, -((b*x^2)/a)])/(4*a^3*b*(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 {x^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac {(A b-a B) x^{1+m}}{4 a b \left (a+b x^2\right )^2}+\frac {(-A b (-3+m)+a B (1+m)) \int \frac {x^m}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {(A b-a B) x^{1+m}}{4 a b \left (a+b x^2\right )^2}+\frac {(A b (3-m)+a B (1+m)) x^{1+m} \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{4 a^3 b (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 80, normalized size = 0.86 \begin {gather*} \frac {x^{1+m} \left (a B \, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )+(A b-a B) \, _2F_1\left (3,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )\right )}{a^3 b (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^m*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(x^(1 + m)*(a*B*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + (A*b - a*B)*Hypergeometric2F1[3, (1
+ m)/2, (3 + m)/2, -((b*x^2)/a)]))/(a^3*b*(1 + m))

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m} \left (B \,x^{2}+A \right )}{\left (b \,x^{2}+a \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(B*x^2+A)/(b*x^2+a)^3,x)

[Out]

int(x^m*(B*x^2+A)/(b*x^2+a)^3,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(x^m*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*x^m/(b*x^2 + a)^3, 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(x^m*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

integral((B*x^2 + A)*x^m/(b^3*x^6 + 3*a*b^2*x^4 + 3*a^2*b*x^2 + a^3), x)

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Sympy [C] Result contains complex when optimal does not.
time = 52.14, size = 3053, normalized size = 32.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

A*(a**2*m**3*x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2
) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 3*a**2*m**2*x*x**m*lerchphi(b*x**2
*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2)
+ 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 2*a**2*m**2*x*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a*
*4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - a**2*m*x*x**m*lerchphi(b*x**2*exp_polar(I*p
i)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**
2*x**4*gamma(m/2 + 3/2)) + 8*a**2*m*x*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m
/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 3*a**2*x*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/
2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2
+ 3/2)) + 10*a**2*x*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**
3*b**2*x**4*gamma(m/2 + 3/2)) + 2*a*b*m**3*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/
2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 6
*a*b*m**2*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2
) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 2*a*b*m**2*x**3*x**m*gamma(m/2 + 1
/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 2*a*b*
m*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a
**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 4*a*b*m*x**3*x**m*gamma(m/2 + 1/2)/(32*a**
5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 6*a*b*x**3*x**m*l
erchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*ga
mma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 6*a*b*x**3*x**m*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3
/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + b**2*m**3*x**5*x**m*lerchphi(b*x
**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3
/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) - 3*b**2*m**2*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 +
1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/
2 + 3/2)) - b**2*m*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(
m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2 + 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2)) + 3*b**2*x**5*x**m*lerchphi
(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 1/2)*gamma(m/2 + 1/2)/(32*a**5*gamma(m/2 + 3/2) + 64*a**4*b*x**2*gamma(m/2
+ 3/2) + 32*a**3*b**2*x**4*gamma(m/2 + 3/2))) + B*(a**2*m**3*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1,
m/2 + 3/2)*gamma(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/2) + 32*a**3*b**2*x**4*ga
mma(m/2 + 5/2)) + 3*a**2*m**2*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*
a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/2) + 32*a**3*b**2*x**4*gamma(m/2 + 5/2)) - 2*a**2*m**2*x*
*3*x**m*gamma(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/2) + 32*a**3*b**2*x**4*gamma
(m/2 + 5/2)) - a**2*m*x**3*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*a**5*gam
ma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/2) + 32*a**3*b**2*x**4*gamma(m/2 + 5/2)) - 3*a**2*x**3*x**m*lerch
phi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(
m/2 + 5/2) + 32*a**3*b**2*x**4*gamma(m/2 + 5/2)) + 18*a**2*x**3*x**m*gamma(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2
) + 64*a**4*b*x**2*gamma(m/2 + 5/2) + 32*a**3*b**2*x**4*gamma(m/2 + 5/2)) + 2*a*b*m**3*x**5*x**m*lerchphi(b*x*
*2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/
2) + 32*a**3*b**2*x**4*gamma(m/2 + 5/2)) + 6*a*b*m**2*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/
2)*gamma(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/2) + 32*a**3*b**2*x**4*gamma(m/2
+ 5/2)) - 2*a*b*m**2*x**5*x**m*gamma(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/2) +
32*a**3*b**2*x**4*gamma(m/2 + 5/2)) - 2*a*b*m*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma
(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/2) + 32*a**3*b**2*x**4*gamma(m/2 + 5/2))
- 4*a*b*m*x**5*x**m*gamma(m/2 + 3/2)/(32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma(m/2 + 5/2) + 32*a**3*b**
2*x**4*gamma(m/2 + 5/2)) - 6*a*b*x**5*x**m*lerchphi(b*x**2*exp_polar(I*pi)/a, 1, m/2 + 3/2)*gamma(m/2 + 3/2)/(
32*a**5*gamma(m/2 + 5/2) + 64*a**4*b*x**2*gamma...

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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(x^m*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*x^m/(b*x^2 + a)^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m\,\left (B\,x^2+A\right )}{{\left (b\,x^2+a\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*(A + B*x^2))/(a + b*x^2)^3,x)

[Out]

int((x^m*(A + B*x^2))/(a + b*x^2)^3, x)

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