3.256 \(\int (d x)^m \left (a+b x^3+c x^6\right )^p \, dx\)

Optimal. Leaf size=155 \[ \frac{(d x)^{m+1} \left (\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (\frac{m+1}{3};-p,-p;\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.278161, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(d x)^{m+1} \left (\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^3+c x^6\right )^p F_1\left (\frac{m+1}{3};-p,-p;\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(d*x)^m*(a + b*x^3 + c*x^6)^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 33.8801, size = 129, normalized size = 0.83 \[ \frac{\left (d x\right )^{m + 1} \left (\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1\right )^{- p} \left (a + b x^{3} + c x^{6}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{m}{3} + \frac{1}{3},- p,- p,\frac{m}{3} + \frac{4}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{d \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*x)**(m + 1)*(2*c*x**3/(b - sqrt(-4*a*c + b**2)) + 1)**(-p)*(2*c*x**3/(b + sqr
t(-4*a*c + b**2)) + 1)**(-p)*(a + b*x**3 + c*x**6)**p*appellf1(m/3 + 1/3, -p, -p
, m/3 + 4/3, -2*c*x**3/(b - sqrt(-4*a*c + b**2)), -2*c*x**3/(b + sqrt(-4*a*c + b
**2)))/(d*(m + 1))

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Mathematica [B]  time = 4.33275, size = 501, normalized size = 3.23 \[ \frac{c (m+4) 2^{-p-1} x \left (\sqrt{b^2-4 a c}+b\right ) (d x)^m \left (x^3 \left (\sqrt{b^2-4 a c}-b\right )-2 a\right )^2 \left (\frac{b-\sqrt{b^2-4 a c}}{2 c}+x^3\right )^{-p} \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^3}{c}\right )^{p+1} \left (a+b x^3+c x^6\right )^{p-1} F_1\left (\frac{m+1}{3};-p,-p;\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{(m+1) \left (\sqrt{b^2-4 a c}-b\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (3 p x^3 \left (\left (\sqrt{b^2-4 a c}-b\right ) F_1\left (\frac{m+4}{3};1-p,-p;\frac{m+7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )-\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+4}{3};-p,1-p;\frac{m+7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )-2 a (m+4) F_1\left (\frac{m+1}{3};-p,-p;\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^6+b*x^3+a)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*(d*x)^m,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{6} + b x^{3} + a\right )}^{p} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*(d*x)^m,x, algorithm="fricas")

[Out]

integral((c*x^6 + b*x^3 + a)^p*(d*x)^m, x)

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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)**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^p*(d*x)^m,x, algorithm="giac")

[Out]

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