Integrand size = 24, antiderivative size = 128 \[ \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx=\frac {2 B (c x)^{1+m} \left (a+b x^n\right )^{5/2}}{b c (2+2 m+5 n)}+\frac {a \left (\frac {A}{1+m}-\frac {2 a B}{b (2+2 m+5 n)}\right ) (c x)^{1+m} \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{c \sqrt {1+\frac {b x^n}{a}}} \] Output:
2*B*(c*x)^(1+m)*(a+b*x^n)^(5/2)/b/c/(2+2*m+5*n)+a*(A/(1+m)-2*a*B/b/(2+2*m+ 5*n))*(c*x)^(1+m)*(a+b*x^n)^(1/2)*hypergeom([-3/2, (1+m)/n],[(1+m+n)/n],-b *x^n/a)/c/(1+b*x^n/a)^(1/2)
Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.92 \[ \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx=\frac {a x (c x)^m \sqrt {a+b x^n} \left (A (1+m+n) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )+B (1+m) x^n \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m+n}{n},\frac {1+m+2 n}{n},-\frac {b x^n}{a}\right )\right )}{(1+m) (1+m+n) \sqrt {1+\frac {b x^n}{a}}} \] Input:
Integrate[(c*x)^m*(a + b*x^n)^(3/2)*(A + B*x^n),x]
Output:
(a*x*(c*x)^m*Sqrt[a + b*x^n]*(A*(1 + m + n)*Hypergeometric2F1[-3/2, (1 + m )/n, (1 + m + n)/n, -((b*x^n)/a)] + B*(1 + m)*x^n*Hypergeometric2F1[-3/2, (1 + m + n)/n, (1 + m + 2*n)/n, -((b*x^n)/a)]))/((1 + m)*(1 + m + n)*Sqrt[ 1 + (b*x^n)/a])
Time = 0.48 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.11, 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.
| \(\displaystyle \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx\) |
| \(\Big \downarrow \) 959 |
| \(\displaystyle \frac {2 B (c x)^{m+1} \left (a+b x^n\right )^{5/2}}{b c (2 m+5 n+2)}-\frac {(2 a B (m+1)-A b (2 m+5 n+2)) \int (c x)^m \left (b x^n+a\right )^{3/2}dx}{b (2 m+5 n+2)}\) |
| \(\Big \downarrow \) 889 |
| \(\displaystyle \frac {2 B (c x)^{m+1} \left (a+b x^n\right )^{5/2}}{b c (2 m+5 n+2)}-\frac {a \sqrt {a+b x^n} (2 a B (m+1)-A b (2 m+5 n+2)) \int (c x)^m \left (\frac {b x^n}{a}+1\right )^{3/2}dx}{b (2 m+5 n+2) \sqrt {\frac {b x^n}{a}+1}}\) |
| \(\Big \downarrow \) 888 |
| \(\displaystyle \frac {2 B (c x)^{m+1} \left (a+b x^n\right )^{5/2}}{b c (2 m+5 n+2)}-\frac {a (c x)^{m+1} \sqrt {a+b x^n} (2 a B (m+1)-A b (2 m+5 n+2)) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{b c (m+1) (2 m+5 n+2) \sqrt {\frac {b x^n}{a}+1}}\) |
Input:
Int[(c*x)^m*(a + b*x^n)^(3/2)*(A + B*x^n),x]
Output:
(2*B*(c*x)^(1 + m)*(a + b*x^n)^(5/2))/(b*c*(2 + 2*m + 5*n)) - (a*(2*a*B*(1 + m) - A*b*(2 + 2*m + 5*n))*(c*x)^(1 + m)*Sqrt[a + b*x^n]*Hypergeometric2 F1[-3/2, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(b*c*(1 + m)*(2 + 2*m + 5*n)*Sqrt[1 + (b*x^n)/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]
\[\int \left (c x \right )^{m} \left (a +b \,x^{n}\right )^{\frac {3}{2}} \left (A +B \,x^{n}\right )d x\]
Input:
int((c*x)^m*(a+b*x^n)^(3/2)*(A+B*x^n),x)
Output:
int((c*x)^m*(a+b*x^n)^(3/2)*(A+B*x^n),x)
Exception generated. \[ \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c*x)^m*(a+b*x^n)^(3/2)*(A+B*x^n),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Result contains complex when optimal does not.
Time = 23.22 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.75 \[ \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx=\frac {A a a^{\frac {m}{n} + \frac {1}{n}} a^{- \frac {m}{n} + \frac {1}{2} - \frac {1}{n}} c^{m} x^{m + 1} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {A a^{- \frac {m}{n} - \frac {1}{2} - \frac {1}{n}} a^{\frac {m}{n} + 1 + \frac {1}{n}} b c^{m} x^{m + n + 1} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{n} + 1 + \frac {1}{n} \\ \frac {m}{n} + 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B a a^{- \frac {m}{n} - \frac {1}{2} - \frac {1}{n}} a^{\frac {m}{n} + 1 + \frac {1}{n}} c^{m} x^{m + n + 1} \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{n} + 1 + \frac {1}{n} \\ \frac {m}{n} + 2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right )} + \frac {B a^{- \frac {m}{n} - \frac {3}{2} - \frac {1}{n}} a^{\frac {m}{n} + 2 + \frac {1}{n}} b c^{m} x^{m + 2 n + 1} \Gamma \left (\frac {m}{n} + 2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{n} + 2 + \frac {1}{n} \\ \frac {m}{n} + 3 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 3 + \frac {1}{n}\right )} \] Input:
integrate((c*x)**m*(a+b*x**n)**(3/2)*(A+B*x**n),x)
Output:
A*a*a**(m/n + 1/n)*a**(-m/n + 1/2 - 1/n)*c**m*x**(m + 1)*gamma(m/n + 1/n)* hyper((-1/2, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*ga mma(m/n + 1 + 1/n)) + A*a**(-m/n - 1/2 - 1/n)*a**(m/n + 1 + 1/n)*b*c**m*x* *(m + n + 1)*gamma(m/n + 1 + 1/n)*hyper((-1/2, m/n + 1 + 1/n), (m/n + 2 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 2 + 1/n)) + B*a*a**(-m/n - 1/2 - 1/n)*a**(m/n + 1 + 1/n)*c**m*x**(m + n + 1)*gamma(m/n + 1 + 1/n)*hy per((-1/2, m/n + 1 + 1/n), (m/n + 2 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n* gamma(m/n + 2 + 1/n)) + B*a**(-m/n - 3/2 - 1/n)*a**(m/n + 2 + 1/n)*b*c**m* x**(m + 2*n + 1)*gamma(m/n + 2 + 1/n)*hyper((-1/2, m/n + 2 + 1/n), (m/n + 3 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 3 + 1/n))
\[ \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx=\int { {\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {3}{2}} \left (c x\right )^{m} \,d x } \] Input:
integrate((c*x)^m*(a+b*x^n)^(3/2)*(A+B*x^n),x, algorithm="maxima")
Output:
integrate((B*x^n + A)*(b*x^n + a)^(3/2)*(c*x)^m, x)
\[ \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx=\int { {\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{\frac {3}{2}} \left (c x\right )^{m} \,d x } \] Input:
integrate((c*x)^m*(a+b*x^n)^(3/2)*(A+B*x^n),x, algorithm="giac")
Output:
integrate((B*x^n + A)*(b*x^n + a)^(3/2)*(c*x)^m, x)
Timed out. \[ \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx=\int {\left (c\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^{3/2} \,d x \] Input:
int((c*x)^m*(A + B*x^n)*(a + b*x^n)^(3/2),x)
Output:
int((c*x)^m*(A + B*x^n)*(a + b*x^n)^(3/2), x)
\[ \int (c x)^m \left (a+b x^n\right )^{3/2} \left (A+B x^n\right ) \, dx=\text {too large to display} \] Input:
int((c*x)^m*(a+b*x^n)^(3/2)*(A+B*x^n),x)
Output:
(c**m*(8*x**(m + 2*n)*sqrt(x**n*b + a)*b**2*m**2*x + 16*x**(m + 2*n)*sqrt( x**n*b + a)*b**2*m*n*x + 16*x**(m + 2*n)*sqrt(x**n*b + a)*b**2*m*x + 6*x** (m + 2*n)*sqrt(x**n*b + a)*b**2*n**2*x + 16*x**(m + 2*n)*sqrt(x**n*b + a)* b**2*n*x + 8*x**(m + 2*n)*sqrt(x**n*b + a)*b**2*x + 16*x**(m + n)*sqrt(x** n*b + a)*a*b*m**2*x + 52*x**(m + n)*sqrt(x**n*b + a)*a*b*m*n*x + 32*x**(m + n)*sqrt(x**n*b + a)*a*b*m*x + 22*x**(m + n)*sqrt(x**n*b + a)*a*b*n**2*x + 52*x**(m + n)*sqrt(x**n*b + a)*a*b*n*x + 16*x**(m + n)*sqrt(x**n*b + a)* a*b*x + 8*x**m*sqrt(x**n*b + a)*a**2*m**2*x + 36*x**m*sqrt(x**n*b + a)*a** 2*m*n*x + 16*x**m*sqrt(x**n*b + a)*a**2*m*x + 46*x**m*sqrt(x**n*b + a)*a** 2*n**2*x + 36*x**m*sqrt(x**n*b + a)*a**2*n*x + 8*x**m*sqrt(x**n*b + a)*a** 2*x + 120*int((x**m*sqrt(x**n*b + a))/(8*x**n*b*m**3 + 36*x**n*b*m**2*n + 24*x**n*b*m**2 + 46*x**n*b*m*n**2 + 72*x**n*b*m*n + 24*x**n*b*m + 15*x**n* b*n**3 + 46*x**n*b*n**2 + 36*x**n*b*n + 8*x**n*b + 8*a*m**3 + 36*a*m**2*n + 24*a*m**2 + 46*a*m*n**2 + 72*a*m*n + 24*a*m + 15*a*n**3 + 46*a*n**2 + 36 *a*n + 8*a),x)*a**3*m**3*n**3 + 540*int((x**m*sqrt(x**n*b + a))/(8*x**n*b* m**3 + 36*x**n*b*m**2*n + 24*x**n*b*m**2 + 46*x**n*b*m*n**2 + 72*x**n*b*m* n + 24*x**n*b*m + 15*x**n*b*n**3 + 46*x**n*b*n**2 + 36*x**n*b*n + 8*x**n*b + 8*a*m**3 + 36*a*m**2*n + 24*a*m**2 + 46*a*m*n**2 + 72*a*m*n + 24*a*m + 15*a*n**3 + 46*a*n**2 + 36*a*n + 8*a),x)*a**3*m**2*n**4 + 360*int((x**m*sq rt(x**n*b + a))/(8*x**n*b*m**3 + 36*x**n*b*m**2*n + 24*x**n*b*m**2 + 46...