3.129 \(\int \frac {x^m (A+B x^3)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=93 \[ \frac {x^{m+1} (a B (m+1)+A b (5-m)) \, _2F_1\left (2,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{6 a^3 b (m+1)}+\frac {x^{m+1} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

1/6*(A*b-B*a)*x^(1+m)/a/b/(b*x^3+a)^2+1/6*(A*b*(5-m)+a*B*(1+m))*x^(1+m)*hypergeom([2, 1/3+1/3*m],[4/3+1/3*m],-
b*x^3/a)/a^3/b/(1+m)

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Rubi [A]  time = 0.04, 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 = {457, 364} \[ \frac {x^{m+1} (a B (m+1)+A b (5-m)) \, _2F_1\left (2,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )}{6 a^3 b (m+1)}+\frac {x^{m+1} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*x^(1 + m))/(6*a*b*(a + b*x^3)^2) + ((A*b*(5 - m) + a*B*(1 + m))*x^(1 + m)*Hypergeometric2F1[2, (1
+ m)/3, (4 + m)/3, -((b*x^3)/a)])/(6*a^3*b*(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 457

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^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {(A b-a B) x^{1+m}}{6 a b \left (a+b x^3\right )^2}+\frac {(-A b (-5+m)+a B (1+m)) \int \frac {x^m}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {(A b-a B) x^{1+m}}{6 a b \left (a+b x^3\right )^2}+\frac {(A b (5-m)+a B (1+m)) x^{1+m} \, _2F_1\left (2,\frac {1+m}{3};\frac {4+m}{3};-\frac {b x^3}{a}\right )}{6 a^3 b (1+m)}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 80, normalized size = 0.86 \[ \frac {x^{m+1} \left ((A b-a B) \, _2F_1\left (3,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )+a B \, _2F_1\left (2,\frac {m+1}{3};\frac {m+4}{3};-\frac {b x^3}{a}\right )\right )}{a^3 b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*(a*B*Hypergeometric2F1[2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)] + (A*b - a*B)*Hypergeometric2F1[3, (1
+ m)/3, (4 + m)/3, -((b*x^3)/a)]))/(a^3*b*(1 + m))

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fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{3} + A\right )} x^{m}}{b^{3} x^{9} + 3 \, a b^{2} x^{6} + 3 \, a^{2} b x^{3} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*x^3 + A)*x^m/(b^3*x^9 + 3*a*b^2*x^6 + 3*a^2*b*x^3 + a^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{3}+A \right ) x^{m}}{\left (b \,x^{3}+a \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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