3.251 \(\int \frac{(d x)^m}{(a+b x^3+c x^6)^2} \, dx\)

Optimal. Leaf size=315 \[ \frac{c (d x)^{m+1} \left (b (2-m) \sqrt{b^2-4 a c}-4 a c (5-m)+b^2 (2-m)\right ) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}\right )}{3 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (-b (2-m) \sqrt{b^2-4 a c}-4 a c (5-m)+b^2 (2-m)\right ) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{3 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^3\right )}{3 a d \left (b^2-4 a c\right ) \left (a+b x^3+c x^6\right )} \]

[Out]

((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*d*(a + b*x^3 + c*x^6)) + (c*(b^2*(2 - m) + b*Sqrt[b
^2 - 4*a*c]*(2 - m) - 4*a*c*(5 - m))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, (-2*c*x^3)/(b -
Sqrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*d*(1 + m)) - (c*(b^2*(2 - m) - b*Sqrt[b^
2 - 4*a*c]*(2 - m) - 4*a*c*(5 - m))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, (-2*c*x^3)/(b + S
qrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*d*(1 + m))

________________________________________________________________________________________

Rubi [A]  time = 0.713023, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1366, 1510, 364} \[ \frac{c (d x)^{m+1} \left (b (2-m) \sqrt{b^2-4 a c}-4 a c (5-m)+b^2 (2-m)\right ) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}\right )}{3 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (-b (2-m) \sqrt{b^2-4 a c}-4 a c (5-m)+b^2 (2-m)\right ) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{3 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^3\right )}{3 a d \left (b^2-4 a c\right ) \left (a+b x^3+c x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m/(a + b*x^3 + c*x^6)^2,x]

[Out]

((d*x)^(1 + m)*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*d*(a + b*x^3 + c*x^6)) + (c*(b^2*(2 - m) + b*Sqrt[b
^2 - 4*a*c]*(2 - m) - 4*a*c*(5 - m))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, (-2*c*x^3)/(b -
Sqrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)^(3/2)*(b - Sqrt[b^2 - 4*a*c])*d*(1 + m)) - (c*(b^2*(2 - m) - b*Sqrt[b^
2 - 4*a*c]*(2 - m) - 4*a*c*(5 - m))*(d*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/3, (4 + m)/3, (-2*c*x^3)/(b + S
qrt[b^2 - 4*a*c])])/(3*a*(b^2 - 4*a*c)^(3/2)*(b + Sqrt[b^2 - 4*a*c])*d*(1 + m))

Rule 1366

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((d*x)^(m + 1)*(b
^2 - 2*a*c + b*c*x^n)*(a + b*x^n + c*x^(2*n))^(p + 1))/(a*d*n*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(a*n*(p + 1)
*(b^2 - 4*a*c)), Int[(d*x)^m*(a + b*x^n + c*x^(2*n))^(p + 1)*Simp[b^2*(m + n*(p + 1) + 1) - 2*a*c*(m + 2*n*(p
+ 1) + 1) + b*c*(m + n*(2*p + 3) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[n2, 2*n] && NeQ[b^2
- 4*a*c, 0] && IGtQ[n, 0] && ILtQ[p, -1]

Rule 1510

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_)))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[e/2 + (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 - q/2 + c*x^n), x], x] + Dist[e/
2 - (2*c*d - b*e)/(2*q), Int[(f*x)^m/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[n2
, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]

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])

Rubi steps

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

Mathematica [C]  time = 0.0783945, size = 78, normalized size = 0.25 \[ \frac{x (d x)^m F_1\left (\frac{m+1}{3};2,2;\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{a^2 (m+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*x)^m/(a + b*x^3 + c*x^6)^2,x]

[Out]

(x*(d*x)^m*AppellF1[(1 + m)/3, 2, 2, (4 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 -
4*a*c])])/(a^2*(1 + m))

________________________________________________________________________________________

Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m/(c*x^6+b*x^3+a)^2,x)

[Out]

int((d*x)^m/(c*x^6+b*x^3+a)^2,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{{\left (c x^{6} + b x^{3} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(c*x^6+b*x^3+a)^2,x, algorithm="maxima")

[Out]

integrate((d*x)^m/(c*x^6 + b*x^3 + a)^2, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (d x\right )^{m}}{c^{2} x^{12} + 2 \, b c x^{9} +{\left (b^{2} + 2 \, a c\right )} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(c*x^6+b*x^3+a)^2,x, algorithm="fricas")

[Out]

integral((d*x)^m/(c^2*x^12 + 2*b*c*x^9 + (b^2 + 2*a*c)*x^6 + 2*a*b*x^3 + a^2), x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m/(c*x**6+b*x**3+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m}}{{\left (c x^{6} + b x^{3} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m/(c*x^6+b*x^3+a)^2,x, algorithm="giac")

[Out]

integrate((d*x)^m/(c*x^6 + b*x^3 + a)^2, x)