3.253 \(\int (d x)^m \sqrt{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=157 \[ \frac{(d x)^{m+1} \sqrt{a+b x^3+c x^6} F_1\left (\frac{m+1}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

[Out]

((d*x)^(1 + m)*Sqrt[a + b*x^3 + c*x^6]*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])])/(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 = 0.449134, antiderivative size = 157, 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 \[ \frac{(d x)^{m+1} \sqrt{a+b x^3+c x^6} F_1\left (\frac{m+1}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \sqrt{\frac{2 c x^3}{b-\sqrt{b^2-4 a c}}+1} \sqrt{\frac{2 c x^3}{\sqrt{b^2-4 a c}+b}+1}} \]

Antiderivative was successfully verified.

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

[Out]

((d*x)^(1 + m)*Sqrt[a + b*x^3 + c*x^6]*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])])/(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 = 35.3433, size = 138, normalized size = 0.88 \[ \frac{\left (d x\right )^{m + 1} \sqrt{a + b x^{3} + c x^{6}} \operatorname{appellf_{1}}{\left (\frac{m}{3} + \frac{1}{3},- \frac{1}{2},- \frac{1}{2},\frac{m}{3} + \frac{4}{3},- \frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}},- \frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} \right )}}{d \left (m + 1\right ) \sqrt{\frac{2 c x^{3}}{b - \sqrt{- 4 a c + b^{2}}} + 1} \sqrt{\frac{2 c x^{3}}{b + \sqrt{- 4 a c + b^{2}}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x)**m*(c*x**6+b*x**3+a)**(1/2),x)

[Out]

(d*x)**(m + 1)*sqrt(a + b*x**3 + c*x**6)*appellf1(m/3 + 1/3, -1/2, -1/2, 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 = 0.289458, size = 424, normalized size = 2.7 \[ \frac{(m+4) x \left (b-\sqrt{b^2-4 a c}\right ) \left (\sqrt{b^2-4 a c}+b\right ) (d x)^m \left (-\sqrt{b^2-4 a c}+b+2 c x^3\right ) \left (\sqrt{b^2-4 a c}+b+2 c x^3\right ) F_1\left (\frac{m+1}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )}{4 c^2 (m+1) \sqrt{a+b x^3+c x^6} \left (3 x^3 \left (\left (\sqrt{b^2-4 a c}+b\right ) F_1\left (\frac{m+4}{3};-\frac{1}{2},\frac{1}{2};\frac{m+7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )+\left (b-\sqrt{b^2-4 a c}\right ) F_1\left (\frac{m+4}{3};\frac{1}{2},-\frac{1}{2};\frac{m+7}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )+4 a (m+4) F_1\left (\frac{m+1}{3};-\frac{1}{2},-\frac{1}{2};\frac{m+4}{3};-\frac{2 c x^3}{b+\sqrt{b^2-4 a c}},\frac{2 c x^3}{\sqrt{b^2-4 a c}-b}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \[ \int \left ( dx \right ) ^{m}\sqrt{c{x}^{6}+b{x}^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int((d*x)^m*(c*x^6+b*x^3+a)^(1/2),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x)**m*(c*x**6+b*x**3+a)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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