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

Optimal. Leaf size=158 \[ \frac{a (d x)^{m+1} \sqrt{a+b x^3+c x^6} F_1\left (\frac{m+1}{3};-\frac{3}{2},-\frac{3}{2};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

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

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Rubi [A]  time = 0.464182, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a (d x)^{m+1} \sqrt{a+b x^3+c x^6} F_1\left (\frac{m+1}{3};-\frac{3}{2},-\frac{3}{2};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.4125, size = 139, normalized size = 0.88 \[ \frac{a \left (d x\right )^{m + 1} \sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (\frac{m}{3} + \frac{1}{3},- \frac{3}{2},- \frac{3}{2},\frac{m}{3} + \frac{4}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{d \left (m + 1\right ) \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 10.5565, size = 1083, normalized size = 6.85 \[ \frac{\left (b-\sqrt{b^2-4 a c}\right ) \left (b+\sqrt{b^2-4 a c}\right ) x (d x)^m \left (2 c x^3+b-\sqrt{b^2-4 a c}\right ) \left (2 c x^3+b+\sqrt{b^2-4 a c}\right ) \left (\frac{c (m+10) F_1\left (\frac{m+7}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+10}{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 ) x^6}{(m+7) \left (3 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+10}{3};-\frac{1}{2},\frac{1}{2};\frac{m+13}{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+10}{3};\frac{1}{2},-\frac{1}{2};\frac{m+13}{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 ) x^3+4 a (m+10) F_1\left (\frac{m+7}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+10}{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 )}+\frac{b (m+7) F_1\left (\frac{m+4}{3};-\frac{1}{2},-\frac{1}{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 ) x^3}{(m+4) \left (3 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+7}{3};-\frac{1}{2},\frac{1}{2};\frac{m+10}{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+7}{3};\frac{1}{2},-\frac{1}{2};\frac{m+10}{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 ) x^3+4 a (m+7) F_1\left (\frac{m+4}{3};-\frac{1}{2},-\frac{1}{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 )}+\frac{a (m+4) F_1\left (\frac{m+1}{3};-\frac{1}{2},-\frac{1}{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 )}{(m+1) \left (3 \left (\left (b+\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+4}{3};-\frac{1}{2},\frac{1}{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 )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+4}{3};\frac{1}{2},-\frac{1}{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 ) x^3+4 a (m+4) F_1\left (\frac{m+1}{3};-\frac{1}{2},-\frac{1}{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 )\right )}\right )}{4 c^2 \sqrt{c x^6+b x^3+a}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((b - Sqrt[b^2 - 4*a*c])*(b + Sqrt[b^2 - 4*a*c])*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)*((a*(4 + m)*AppellF1[(1 + m)/3, -
1/2, -1/2, (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])])/((1 + m)*(4*a*(4 + m)*AppellF1[(1 + m)/3, -1/2, -1/2, (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, -1/2, 1/2, (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])] + (b - Sqrt[b^2 - 4*a
*c])*AppellF1[(4 + m)/3, 1/2, -1/2, (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])]))) + (b*(7 + m)*x^3*AppellF1[(4 + m)/3, -
1/2, -1/2, (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])])/((4 + m)*(4*a*(7 + m)*AppellF1[(4 + m)/3, -1/2, -1/2, (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])] + 3*x^3*
((b + Sqrt[b^2 - 4*a*c])*AppellF1[(7 + m)/3, -1/2, 1/2, (10 + 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[(7 + m)/3, 1/2, -1/2, (10 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*
c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])]))) + (c*(10 + m)*x^6*AppellF1[(7 + m)/3
, -1/2, -1/2, (10 + m)/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sq
rt[b^2 - 4*a*c])])/((7 + m)*(4*a*(10 + m)*AppellF1[(7 + m)/3, -1/2, -1/2, (10 +
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[(10 + m)/3, -1/2, 1/2, (13 + 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[(10 + m)/3, 1/2, -1/2, (13 + 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^2*Sqrt[a + b*x^3 + c*
x^6])

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Maple [F]  time = 0.014, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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