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

Optimal. Leaf size=131 \[ \frac{2 B \sqrt{a+b x^3} (e x)^{m+1}}{b e (2 m+5)}-\frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} (2 a B (m+1)-A b (2 m+5)) \, _2F_1\left (\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{b e (m+1) (2 m+5) \sqrt{a+b x^3}} \]

[Out]

(2*B*(e*x)^(1 + m)*Sqrt[a + b*x^3])/(b*e*(5 + 2*m)) - ((2*a*B*(1 + m) - A*b*(5 +
2*m))*(e*x)^(1 + m)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, (1 + m)/3, (4 +
m)/3, -((b*x^3)/a)])/(b*e*(1 + m)*(5 + 2*m)*Sqrt[a + b*x^3])

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Rubi [A]  time = 0.243811, 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 \[ \frac{\sqrt{\frac{b x^3}{a}+1} (e x)^{m+1} \left (\frac{A}{m+1}-\frac{2 a B}{2 b m+5 b}\right ) \, _2F_1\left (\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{e \sqrt{a+b x^3}}+\frac{2 B \sqrt{a+b x^3} (e x)^{m+1}}{b e (2 m+5)} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*(e*x)^(1 + m)*Sqrt[a + b*x^3])/(b*e*(5 + 2*m)) + ((A/(1 + m) - (2*a*B)/(5*b
+ 2*b*m))*(e*x)^(1 + m)*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/2, (1 + m)/3, (
4 + m)/3, -((b*x^3)/a)])/(e*Sqrt[a + b*x^3])

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Rubi in Sympy [A]  time = 18.8866, size = 109, normalized size = 0.83 \[ \frac{2 B \left (e x\right )^{m + 1} \sqrt{a + b x^{3}}}{b e \left (2 m + 5\right )} + \frac{\left (e x\right )^{m + 1} \sqrt{a + b x^{3}} \left (A b \left (2 m + 5\right ) - 2 B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{a b e \sqrt{1 + \frac{b x^{3}}{a}} \left (m + 1\right ) \left (2 m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(e*x)**(m + 1)*sqrt(a + b*x**3)/(b*e*(2*m + 5)) + (e*x)**(m + 1)*sqrt(a + b*
x**3)*(A*b*(2*m + 5) - 2*B*a*(m + 1))*hyper((1/2, m/3 + 1/3), (m/3 + 4/3,), -b*x
**3/a)/(a*b*e*sqrt(1 + b*x**3/a)*(m + 1)*(2*m + 5))

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Mathematica [A]  time = 0.142292, size = 110, normalized size = 0.84 \[ \frac{x \sqrt{a+b x^3} (e x)^m \left ((A b-a B) \, _2F_1\left (\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )+a B \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )\right )}{a b (m+1) \sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{ \left ( ex \right ) ^{m} \left ( B{x}^{3}+A \right ){\frac{1}{\sqrt{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 7.47346, size = 119, normalized size = 0.91 \[ \frac{A e^{m} x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} + \frac{B e^{m} x^{4} x^{m} \Gamma \left (\frac{m}{3} + \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{3} + \frac{4}{3} \\ \frac{m}{3} + \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{m}{3} + \frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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*sqrt(a)*gamma(m/3 + 4/3)) + B*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*sqrt(a)*gamma
(m/3 + 7/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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