3.501 \(\int (e x)^m \left (a+b x^3\right )^{3/2} \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=132 \[ \frac{2 B \left (a+b x^3\right )^{5/2} (e x)^{m+1}}{b e (2 m+17)}-\frac{a \sqrt{a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+17)) \, _2F_1\left (-\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{b e (m+1) (2 m+17) \sqrt{\frac{b x^3}{a}+1}} \]

[Out]

(2*B*(e*x)^(1 + m)*(a + b*x^3)^(5/2))/(b*e*(17 + 2*m)) - (a*(2*a*B*(1 + m) - A*b
*(17 + 2*m))*(e*x)^(1 + m)*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, (1 + m)/3, (4
+ m)/3, -((b*x^3)/a)])/(b*e*(1 + m)*(17 + 2*m)*Sqrt[1 + (b*x^3)/a])

_______________________________________________________________________________________

Rubi [A]  time = 0.246178, antiderivative size = 132, 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 \[ \frac{a \sqrt{a+b x^3} (e x)^{m+1} \left (\frac{A}{m+1}-\frac{2 a B}{2 b m+17 b}\right ) \, _2F_1\left (-\frac{3}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{e \sqrt{\frac{b x^3}{a}+1}}+\frac{2 B \left (a+b x^3\right )^{5/2} (e x)^{m+1}}{b e (2 m+17)} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*(e*x)^(1 + m)*(a + b*x^3)^(5/2))/(b*e*(17 + 2*m)) + (a*(A/(1 + m) - (2*a*B)
/(17*b + 2*b*m))*(e*x)^(1 + m)*Sqrt[a + b*x^3]*Hypergeometric2F1[-3/2, (1 + m)/3
, (4 + m)/3, -((b*x^3)/a)])/(e*Sqrt[1 + (b*x^3)/a])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.8444, size = 110, normalized size = 0.83 \[ \frac{2 B \left (e x\right )^{m + 1} \left (a + b x^{3}\right )^{\frac{5}{2}}}{b e \left (2 m + 17\right )} + \frac{a \left (e x\right )^{m + 1} \sqrt{a + b x^{3}} \left (A b \left (2 m + 17\right ) - 2 B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{b e \sqrt{1 + \frac{b x^{3}}{a}} \left (m + 1\right ) \left (2 m + 17\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*(e*x)**(m + 1)*(a + b*x**3)**(5/2)/(b*e*(2*m + 17)) + a*(e*x)**(m + 1)*sqrt(
a + b*x**3)*(A*b*(2*m + 17) - 2*B*a*(m + 1))*hyper((-3/2, m/3 + 1/3), (m/3 + 4/3
,), -b*x**3/a)/(b*e*sqrt(1 + b*x**3/a)*(m + 1)*(2*m + 17))

_______________________________________________________________________________________

Mathematica [A]  time = 0.222564, size = 149, normalized size = 1.13 \[ \frac{x \sqrt{a+b x^3} (e x)^m \left (\frac{x^3 (a B+A b) \, _2F_1\left (-\frac{1}{2},\frac{m+4}{3};\frac{m+7}{3};-\frac{b x^3}{a}\right )}{m+4}+\frac{a A \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{m+1}+\frac{b B x^6 \, _2F_1\left (-\frac{1}{2},\frac{m+7}{3};\frac{m+10}{3};-\frac{b x^3}{a}\right )}{m+7}\right )}{\sqrt{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(e*x)^m*Sqrt[a + b*x^3]*((a*A*Hypergeometric2F1[-1/2, (1 + m)/3, (4 + m)/3, -
((b*x^3)/a)])/(1 + m) + ((A*b + a*B)*x^3*Hypergeometric2F1[-1/2, (4 + m)/3, (7 +
m)/3, -((b*x^3)/a)])/(4 + m) + (b*B*x^6*Hypergeometric2F1[-1/2, (7 + m)/3, (10
+ m)/3, -((b*x^3)/a)])/(7 + m)))/Sqrt[1 + (b*x^3)/a]

_______________________________________________________________________________________

Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( b{x}^{3}+a \right ) ^{{\frac{3}{2}}} \left ( B{x}^{3}+A \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*(e*x)^m,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*(e*x)^m,x, algorithm="fricas")

[Out]

integral((B*b*x^6 + (B*a + A*b)*x^3 + A*a)*sqrt(b*x^3 + a)*(e*x)^m, x)

_______________________________________________________________________________________

Sympy [A]  time = 90.502, size = 252, normalized size = 1.91 \[ \frac{A a^{\frac{3}{2}} e^{m} x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} + \frac{A \sqrt{a} b e^{m} x^{4} x^{m} \Gamma \left (\frac{m}{3} + \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{4}{3} \\ \frac{m}{3} + \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{7}{3}\right )} + \frac{B a^{\frac{3}{2}} e^{m} x^{4} x^{m} \Gamma \left (\frac{m}{3} + \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{4}{3} \\ \frac{m}{3} + \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{7}{3}\right )} + \frac{B \sqrt{a} b e^{m} x^{7} x^{m} \Gamma \left (\frac{m}{3} + \frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{7}{3} \\ \frac{m}{3} + \frac{10}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{10}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**(3/2)*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)) + A*sqrt(a)*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*
gamma(m/3 + 7/3)) + B*a**(3/2)*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*gamma(m/3 + 7/3)) + B*sqrt(a)
*b*e**m*x**7*x**m*gamma(m/3 + 7/3)*hyper((-1/2, m/3 + 7/3), (m/3 + 10/3,), b*x**
3*exp_polar(I*pi)/a)/(3*gamma(m/3 + 10/3))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^(3/2)*(e*x)^m,x, algorithm="giac")

[Out]

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