Integrand size = 24, antiderivative size = 123 \[ \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\frac {2 B (e x)^{1+m} \left (a+b x^3\right )^{7/2}}{b e (23+2 m)}+\frac {a^2 \left (\frac {A}{1+m}-\frac {2 a B}{23 b+2 b m}\right ) (e x)^{1+m} \sqrt {a+b x^3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )}{e \sqrt {1+\frac {b x^3}{a}}} \] Output:
2*B*(e*x)^(1+m)*(b*x^3+a)^(7/2)/b/e/(23+2*m)+a^2*(A/(1+m)-2*a*B/(2*b*m+23* b))*(e*x)^(1+m)*(b*x^3+a)^(1/2)*hypergeom([-5/2, 1/3+1/3*m],[4/3+1/3*m],-b *x^3/a)/e/(1+b*x^3/a)^(1/2)
Time = 0.93 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.92 \[ \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\frac {a^2 x (e x)^m \sqrt {a+b x^3} \left (A (4+m) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{3},\frac {4+m}{3},-\frac {b x^3}{a}\right )+B (1+m) x^3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {4+m}{3},\frac {7+m}{3},-\frac {b x^3}{a}\right )\right )}{(1+m) (4+m) \sqrt {1+\frac {b x^3}{a}}} \] Input:
Integrate[(e*x)^m*(a + b*x^3)^(5/2)*(A + B*x^3),x]
Output:
(a^2*x*(e*x)^m*Sqrt[a + b*x^3]*(A*(4 + m)*Hypergeometric2F1[-5/2, (1 + m)/ 3, (4 + m)/3, -((b*x^3)/a)] + B*(1 + m)*x^3*Hypergeometric2F1[-5/2, (4 + m )/3, (7 + m)/3, -((b*x^3)/a)]))/((1 + m)*(4 + m)*Sqrt[1 + (b*x^3)/a])
Time = 0.46 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {959, 889, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) (e x)^m \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {2 B \left (a+b x^3\right )^{7/2} (e x)^{m+1}}{b e (2 m+23)}-\frac {(2 a B (m+1)-A b (2 m+23)) \int (e x)^m \left (b x^3+a\right )^{5/2}dx}{b (2 m+23)}\) |
| \(\Big \downarrow \) 889 |
| \(\displaystyle \frac {2 B \left (a+b x^3\right )^{7/2} (e x)^{m+1}}{b e (2 m+23)}-\frac {a^2 \sqrt {a+b x^3} (2 a B (m+1)-A b (2 m+23)) \int (e x)^m \left (\frac {b x^3}{a}+1\right )^{5/2}dx}{b (2 m+23) \sqrt {\frac {b x^3}{a}+1}}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {2 B \left (a+b x^3\right )^{7/2} (e x)^{m+1}}{b e (2 m+23)}-\frac {a^2 \sqrt {a+b x^3} (e x)^{m+1} (2 a B (m+1)-A b (2 m+23)) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {m+1}{3},\frac {m+4}{3},-\frac {b x^3}{a}\right )}{b e (m+1) (2 m+23) \sqrt {\frac {b x^3}{a}+1}}\) |
Input:
Int[(e*x)^m*(a + b*x^3)^(5/2)*(A + B*x^3),x]
Output:
(2*B*(e*x)^(1 + m)*(a + b*x^3)^(7/2))/(b*e*(23 + 2*m)) - (a^2*(2*a*B*(1 + m) - A*b*(23 + 2*m))*(e*x)^(1 + m)*Sqrt[a + b*x^3]*Hypergeometric2F1[-5/2, (1 + m)/3, (4 + m)/3, -((b*x^3)/a)])/(b*e*(1 + m)*(23 + 2*m)*Sqrt[1 + (b* x^3)/a])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
\[\int \left (e x \right )^{m} \left (b \,x^{3}+a \right )^{\frac {5}{2}} \left (B \,x^{3}+A \right )d x\]
Input:
int((e*x)^m*(b*x^3+a)^(5/2)*(B*x^3+A),x)
Output:
int((e*x)^m*(b*x^3+a)^(5/2)*(B*x^3+A),x)
\[ \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(b*x^3+a)^(5/2)*(B*x^3+A),x, algorithm="fricas")
Output:
integral((B*b^2*x^9 + (2*B*a*b + A*b^2)*x^6 + (B*a^2 + 2*A*a*b)*x^3 + A*a^ 2)*sqrt(b*x^3 + a)*(e*x)^m, x)
Result contains complex when optimal does not.
Time = 20.41 (sec) , antiderivative size = 379, normalized size of antiderivative = 3.08 \[ \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\frac {A a^{\frac {5}{2}} e^{m} x^{m + 1} \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 {2 A a^{\frac {3}{2}} b e^{m} x^{m + 4} \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 {A \sqrt {a} b^{2} e^{m} x^{m + 7} \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 )} + \frac {B a^{\frac {5}{2}} e^{m} x^{m + 4} \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 {2 B a^{\frac {3}{2}} b e^{m} x^{m + 7} \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 )} + \frac {B \sqrt {a} b^{2} e^{m} x^{m + 10} \Gamma \left (\frac {m}{3} + \frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{3} + \frac {10}{3} \\ \frac {m}{3} + \frac {13}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {m}{3} + \frac {13}{3}\right )} \] Input:
integrate((e*x)**m*(b*x**3+a)**(5/2)*(B*x**3+A),x)
Output:
A*a**(5/2)*e**m*x**(m + 1)*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)) + 2*A*a**(3/2)*b*e **m*x**(m + 4)*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)) + A*sqrt(a)*b**2*e**m*x**(m + 7)*gamma(m/3 + 7/3)*hyper((-1/2, m/3 + 7/3), (m/3 + 10/3,), b*x**3*exp_pol ar(I*pi)/a)/(3*gamma(m/3 + 10/3)) + B*a**(5/2)*e**m*x**(m + 4)*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)) + 2*B*a**(3/2)*b*e**m*x**(m + 7)*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)) + B*sqrt(a)*b**2*e**m*x**(m + 10)*gamma(m/3 + 10/3)*hyper((-1/2, m /3 + 10/3), (m/3 + 13/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(m/3 + 13/3))
\[ \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(b*x^3+a)^(5/2)*(B*x^3+A),x, algorithm="maxima")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^m, x)
\[ \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\int { {\left (B x^{3} + A\right )} {\left (b x^{3} + a\right )}^{\frac {5}{2}} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(b*x^3+a)^(5/2)*(B*x^3+A),x, algorithm="giac")
Output:
integrate((B*x^3 + A)*(b*x^3 + a)^(5/2)*(e*x)^m, x)
Timed out. \[ \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\int \left (B\,x^3+A\right )\,{\left (e\,x\right )}^m\,{\left (b\,x^3+a\right )}^{5/2} \,d x \] Input:
int((A + B*x^3)*(e*x)^m*(a + b*x^3)^(5/2),x)
Output:
int((A + B*x^3)*(e*x)^m*(a + b*x^3)^(5/2), x)
\[ \int (e x)^m \left (a+b x^3\right )^{5/2} \left (A+B x^3\right ) \, dx=\frac {e^{m} \left (16 x^{m} \sqrt {b \,x^{3}+a}\, a^{3} m^{3} x +432 x^{m} \sqrt {b \,x^{3}+a}\, a^{3} m^{2} x +3912 x^{m} \sqrt {b \,x^{3}+a}\, a^{3} m x +13000 x^{m} \sqrt {b \,x^{3}+a}\, a^{3} x +48 x^{m} \sqrt {b \,x^{3}+a}\, a^{2} b \,m^{3} x^{4}+1128 x^{m} \sqrt {b \,x^{3}+a}\, a^{2} b \,m^{2} x^{4}+7872 x^{m} \sqrt {b \,x^{3}+a}\, a^{2} b m \,x^{4}+13380 x^{m} \sqrt {b \,x^{3}+a}\, a^{2} b \,x^{4}+48 x^{m} \sqrt {b \,x^{3}+a}\, a \,b^{2} m^{3} x^{7}+960 x^{m} \sqrt {b \,x^{3}+a}\, a \,b^{2} m^{2} x^{7}+5268 x^{m} \sqrt {b \,x^{3}+a}\, a \,b^{2} m \,x^{7}+7920 x^{m} \sqrt {b \,x^{3}+a}\, a \,b^{2} x^{7}+16 x^{m} \sqrt {b \,x^{3}+a}\, b^{3} m^{3} x^{10}+264 x^{m} \sqrt {b \,x^{3}+a}\, b^{3} m^{2} x^{10}+1308 x^{m} \sqrt {b \,x^{3}+a}\, b^{3} m \,x^{10}+1870 x^{m} \sqrt {b \,x^{3}+a}\, b^{3} x^{10}+136080 \left (\int \frac {x^{m} \sqrt {b \,x^{3}+a}}{16 b \,m^{4} x^{3}+448 b \,m^{3} x^{3}+4344 b \,m^{2} x^{3}+16 a \,m^{4}+16912 b m \,x^{3}+448 a \,m^{3}+21505 b \,x^{3}+4344 a \,m^{2}+16912 a m +21505 a}d x \right ) a^{4} m^{4}+3810240 \left (\int \frac {x^{m} \sqrt {b \,x^{3}+a}}{16 b \,m^{4} x^{3}+448 b \,m^{3} x^{3}+4344 b \,m^{2} x^{3}+16 a \,m^{4}+16912 b m \,x^{3}+448 a \,m^{3}+21505 b \,x^{3}+4344 a \,m^{2}+16912 a m +21505 a}d x \right ) a^{4} m^{3}+36945720 \left (\int \frac {x^{m} \sqrt {b \,x^{3}+a}}{16 b \,m^{4} x^{3}+448 b \,m^{3} x^{3}+4344 b \,m^{2} x^{3}+16 a \,m^{4}+16912 b m \,x^{3}+448 a \,m^{3}+21505 b \,x^{3}+4344 a \,m^{2}+16912 a m +21505 a}d x \right ) a^{4} m^{2}+143836560 \left (\int \frac {x^{m} \sqrt {b \,x^{3}+a}}{16 b \,m^{4} x^{3}+448 b \,m^{3} x^{3}+4344 b \,m^{2} x^{3}+16 a \,m^{4}+16912 b m \,x^{3}+448 a \,m^{3}+21505 b \,x^{3}+4344 a \,m^{2}+16912 a m +21505 a}d x \right ) a^{4} m +182900025 \left (\int \frac {x^{m} \sqrt {b \,x^{3}+a}}{16 b \,m^{4} x^{3}+448 b \,m^{3} x^{3}+4344 b \,m^{2} x^{3}+16 a \,m^{4}+16912 b m \,x^{3}+448 a \,m^{3}+21505 b \,x^{3}+4344 a \,m^{2}+16912 a m +21505 a}d x \right ) a^{4}\right )}{16 m^{4}+448 m^{3}+4344 m^{2}+16912 m +21505} \] Input:
int((e*x)^m*(b*x^3+a)^(5/2)*(B*x^3+A),x)
Output:
(e**m*(16*x**m*sqrt(a + b*x**3)*a**3*m**3*x + 432*x**m*sqrt(a + b*x**3)*a* *3*m**2*x + 3912*x**m*sqrt(a + b*x**3)*a**3*m*x + 13000*x**m*sqrt(a + b*x* *3)*a**3*x + 48*x**m*sqrt(a + b*x**3)*a**2*b*m**3*x**4 + 1128*x**m*sqrt(a + b*x**3)*a**2*b*m**2*x**4 + 7872*x**m*sqrt(a + b*x**3)*a**2*b*m*x**4 + 13 380*x**m*sqrt(a + b*x**3)*a**2*b*x**4 + 48*x**m*sqrt(a + b*x**3)*a*b**2*m* *3*x**7 + 960*x**m*sqrt(a + b*x**3)*a*b**2*m**2*x**7 + 5268*x**m*sqrt(a + b*x**3)*a*b**2*m*x**7 + 7920*x**m*sqrt(a + b*x**3)*a*b**2*x**7 + 16*x**m*s qrt(a + b*x**3)*b**3*m**3*x**10 + 264*x**m*sqrt(a + b*x**3)*b**3*m**2*x**1 0 + 1308*x**m*sqrt(a + b*x**3)*b**3*m*x**10 + 1870*x**m*sqrt(a + b*x**3)*b **3*x**10 + 136080*int((x**m*sqrt(a + b*x**3))/(16*a*m**4 + 448*a*m**3 + 4 344*a*m**2 + 16912*a*m + 21505*a + 16*b*m**4*x**3 + 448*b*m**3*x**3 + 4344 *b*m**2*x**3 + 16912*b*m*x**3 + 21505*b*x**3),x)*a**4*m**4 + 3810240*int(( x**m*sqrt(a + b*x**3))/(16*a*m**4 + 448*a*m**3 + 4344*a*m**2 + 16912*a*m + 21505*a + 16*b*m**4*x**3 + 448*b*m**3*x**3 + 4344*b*m**2*x**3 + 16912*b*m *x**3 + 21505*b*x**3),x)*a**4*m**3 + 36945720*int((x**m*sqrt(a + b*x**3))/ (16*a*m**4 + 448*a*m**3 + 4344*a*m**2 + 16912*a*m + 21505*a + 16*b*m**4*x* *3 + 448*b*m**3*x**3 + 4344*b*m**2*x**3 + 16912*b*m*x**3 + 21505*b*x**3),x )*a**4*m**2 + 143836560*int((x**m*sqrt(a + b*x**3))/(16*a*m**4 + 448*a*m** 3 + 4344*a*m**2 + 16912*a*m + 21505*a + 16*b*m**4*x**3 + 448*b*m**3*x**3 + 4344*b*m**2*x**3 + 16912*b*m*x**3 + 21505*b*x**3),x)*a**4*m + 18290002...