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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]