3.502 \(\int (e x)^m \sqrt{a+b x^3} \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=131 \[ \frac{2 B \left (a+b x^3\right )^{3/2} (e x)^{m+1}}{b e (2 m+11)}-\frac{\sqrt{a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+11)) \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{b e (m+1) (2 m+11) \sqrt{\frac{b x^3}{a}+1}} \]

[Out]

(2*B*(e*x)^(1 + m)*(a + b*x^3)^(3/2))/(b*e*(11 + 2*m)) - ((2*a*B*(1 + m) - A*b*(
11 + 2*m))*(e*x)^(1 + m)*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, (1 + m)/3, (4 +
m)/3, -((b*x^3)/a)])/(b*e*(1 + m)*(11 + 2*m)*Sqrt[1 + (b*x^3)/a])

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Rubi [A]  time = 0.244327, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{a+b x^3} (e x)^{m+1} \left (\frac{A}{m+1}-\frac{2 a B}{2 b m+11 b}\right ) \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{e \sqrt{\frac{b x^3}{a}+1}}+\frac{2 B \left (a+b x^3\right )^{3/2} (e x)^{m+1}}{b e (2 m+11)} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(2*B*(e*x)^(1 + m)*(a + b*x^3)^(3/2))/(b*e*(11 + 2*m)) + ((A/(1 + m) - (2*a*B)/(
11*b + 2*b*m))*(e*x)^(1 + m)*Sqrt[a + b*x^3]*Hypergeometric2F1[-1/2, (1 + m)/3,
(4 + m)/3, -((b*x^3)/a)])/(e*Sqrt[1 + (b*x^3)/a])

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Rubi in Sympy [A]  time = 17.5889, size = 109, normalized size = 0.83 \[ \frac{2 B \left (e x\right )^{m + 1} \left (a + b x^{3}\right )^{\frac{3}{2}}}{b e \left (2 m + 11\right )} + \frac{\left (e x\right )^{m + 1} \sqrt{a + b x^{3}} \left (A b \left (2 m + 11\right ) - 2 B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{b e \sqrt{1 + \frac{b x^{3}}{a}} \left (m + 1\right ) \left (2 m + 11\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(b*x**3+a)**(1/2)*(B*x**3+A),x)

[Out]

2*B*(e*x)**(m + 1)*(a + b*x**3)**(3/2)/(b*e*(2*m + 11)) + (e*x)**(m + 1)*sqrt(a
+ b*x**3)*(A*b*(2*m + 11) - 2*B*a*(m + 1))*hyper((-1/2, m/3 + 1/3), (m/3 + 4/3,)
, -b*x**3/a)/(b*e*sqrt(1 + b*x**3/a)*(m + 1)*(2*m + 11))

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Mathematica [A]  time = 0.107004, size = 110, normalized size = 0.84 \[ \frac{x \sqrt{a+b x^3} (e x)^m \left (A (m+4) \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+B (m+1) x^3 \, _2F_1\left (-\frac{1}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right )\right )}{(m+1) (m+4) \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m*Sqrt[a + b*x^3]*(A + B*x^3),x]

[Out]

(x*(e*x)^m*Sqrt[a + b*x^3]*(A*(4 + m)*Hypergeometric2F1[-1/2, (1 + m)/3, (4 + m)
/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[-1/2, (4 + m)/3, (7 + m)/3,
-((b*x^3)/a)]))/((1 + m)*(4 + m)*Sqrt[1 + (b*x^3)/a])

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m}\sqrt{b{x}^{3}+a} \left ( B{x}^{3}+A \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(b*x^3+a)^(1/2)*(B*x^3+A),x)

[Out]

int((e*x)^m*(b*x^3+a)^(1/2)*(B*x^3+A),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^m,x, algorithm="maxima")

[Out]

integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^m,x, algorithm="fricas")

[Out]

integral((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^m, x)

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Sympy [A]  time = 11.3943, size = 122, normalized size = 0.93 \[ \frac{A \sqrt{a} e^{m} x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} + \frac{B \sqrt{a} e^{m} x^{4} x^{m} \Gamma \left (\frac{m}{3} + \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{4}{3} \\ \frac{m}{3} + \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(b*x**3+a)**(1/2)*(B*x**3+A),x)

[Out]

A*sqrt(a)*e**m*x*x**m*gamma(m/3 + 1/3)*hyper((-1/2, m/3 + 1/3), (m/3 + 4/3,), b*
x**3*exp_polar(I*pi)/a)/(3*gamma(m/3 + 4/3)) + B*sqrt(a)*e**m*x**4*x**m*gamma(m/
3 + 4/3)*hyper((-1/2, m/3 + 4/3), (m/3 + 7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gam
ma(m/3 + 7/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^m,x, algorithm="giac")

[Out]

integrate((B*x^3 + A)*sqrt(b*x^3 + a)*(e*x)^m, x)