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

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 + 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))

_______________________________________________________________________________________

Rubi [A]  time = 1.51084, 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 \[ \frac{c (d x)^{m+1} \left (b (2-m) \sqrt{b^2-4 a c}-4 a c (5-m)+b^2 (2-m)\right ) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}\right )}{3 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (-b (2-m) \sqrt{b^2-4 a c}-4 a c (5-m)+b^2 (2-m)\right ) \, _2F_1\left (1,\frac{m+1}{3};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{3 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^3\right )}{3 a d \left (b^2-4 a c\right ) \left (a+b x^3+c x^6\right )} \]

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 + 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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 114.466, size = 264, normalized size = 0.84 \[ - \frac{c \left (d x\right )^{m + 1} \left (- 4 a c \left (- m + 5\right ) + b^{2} \left (- m + 2\right ) - b \left (- m + 2\right ) \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{3 a d \left (b + \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{c \left (d x\right )^{m + 1} \left (- 4 a c \left (- m + 5\right ) + b^{2} \left (- m + 2\right ) + b \left (- m + 2\right ) \sqrt{- 4 a c + b^{2}}\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{3 a d \left (b - \sqrt{- 4 a c + b^{2}}\right ) \left (m + 1\right ) \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{\left (d x\right )^{m + 1} \left (- 2 a c + b^{2} + b c x^{3}\right )}{3 a d \left (- 4 a c + b^{2}\right ) \left (a + b x^{3} + c x^{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c*(d*x)**(m + 1)*(-4*a*c*(-m + 5) + b**2*(-m + 2) - b*(-m + 2)*sqrt(-4*a*c + b*
*2))*hyper((1, m/3 + 1/3), (m/3 + 4/3,), -2*c*x**3/(b + sqrt(-4*a*c + b**2)))/(3
*a*d*(b + sqrt(-4*a*c + b**2))*(m + 1)*(-4*a*c + b**2)**(3/2)) + c*(d*x)**(m + 1
)*(-4*a*c*(-m + 5) + b**2*(-m + 2) + b*(-m + 2)*sqrt(-4*a*c + b**2))*hyper((1, m
/3 + 1/3), (m/3 + 4/3,), -2*c*x**3/(b - sqrt(-4*a*c + b**2)))/(3*a*d*(b - sqrt(-
4*a*c + b**2))*(m + 1)*(-4*a*c + b**2)**(3/2)) + (d*x)**(m + 1)*(-2*a*c + b**2 +
b*c*x**3)/(3*a*d*(-4*a*c + b**2)*(a + b*x**3 + c*x**6))

_______________________________________________________________________________________

Mathematica [C]  time = 1.6524, size = 376, normalized size = 1.19 \[ \frac{a (m+4) x (d x)^m \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) F_1\left (\frac{m+1}{3};2,2;\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{4 c (m+1) \left (a+b x^3+c x^6\right )^3 \left (a (m+4) F_1\left (\frac{m+1}{3};2,2;\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-3 x^3 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+4}{3};2,3;\frac{m+7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+4}{3};3,2;\frac{m+7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*(4 + m)*x*(d*x)^m*(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)*(b + Sqrt[b^2 - 4*a*c] +
2*c*x^3)*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])])/(4*c*(1 + m)*(a + b*x^3 + c*x^6)^3*(a*(4
+ 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])] - 3*x^3*((b + Sqrt[b^2 - 4*a*c])*AppellF1[(4 +
m)/3, 2, 3, (7 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt
[b^2 - 4*a*c])] + (b - Sqrt[b^2 - 4*a*c])*AppellF1[(4 + m)/3, 3, 2, (7 + m)/3, (
-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])))

_______________________________________________________________________________________

Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{\frac{ \left ( dx \right ) ^{m}}{ \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{2}}}\, dx \]

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. \[ \int \frac{\left (d x\right )^{m}}{{\left (c x^{6} + b x^{3} + a\right )}^{2}}\,{d x} \]

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. \[{\rm integral}\left (\frac{\left (d x\right )^{m}}{c^{2} x^{12} + 2 \, b c x^{9} +{\left (b^{2} + 2 \, a c\right )} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]

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. \[ \text{Timed out} \]

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

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)