Integrand size = 21, antiderivative size = 107 \[ \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx=\frac {2 b x^{1+n} (c x)^m \sqrt {a x^j+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m+\frac {3 n}{2}}{j-n},1+\frac {1+m+\frac {3 n}{2}}{j-n},-\frac {a x^{j-n}}{b}\right )}{(2+2 m+3 n) \sqrt {1+\frac {a x^{j-n}}{b}}} \] Output:
2*b*x^(1+n)*(c*x)^m*(a*x^j+b*x^n)^(1/2)*hypergeom([-3/2, (1+m+3/2*n)/(j-n) ],[1+(1+m+3/2*n)/(j-n)],-a*x^(j-n)/b)/(2+2*m+3*n)/(1+a*x^(j-n)/b)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(218\) vs. \(2(107)=214\).
Time = 0.41 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.04 \[ \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx=\frac {2 (c x)^m \left ((2+4 j+2 m-n) x^{-m} \left (a x^j+b x^n\right ) \left (a (2-j+2 m+4 n) x^{1+j+m}+b (2+2 j+2 m+n) x^{1+m+n}\right )+3 a^2 (j-n)^2 x^{1+2 j} \sqrt {1+\frac {a x^{j-n}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+4 j+2 m-n}{2 j-2 n},\frac {2+6 j+2 m-3 n}{2 j-2 n},-\frac {a x^{j-n}}{b}\right )\right )}{(2+4 j+2 m-n) (2+2 j+2 m+n) (2+2 m+3 n) \sqrt {a x^j+b x^n}} \] Input:
Integrate[(c*x)^m*(a*x^j + b*x^n)^(3/2),x]
Output:
(2*(c*x)^m*(((2 + 4*j + 2*m - n)*(a*x^j + b*x^n)*(a*(2 - j + 2*m + 4*n)*x^ (1 + j + m) + b*(2 + 2*j + 2*m + n)*x^(1 + m + n)))/x^m + 3*a^2*(j - n)^2* x^(1 + 2*j)*Sqrt[1 + (a*x^(j - n))/b]*Hypergeometric2F1[1/2, (2 + 4*j + 2* m - n)/(2*j - 2*n), (2 + 6*j + 2*m - 3*n)/(2*j - 2*n), -((a*x^(j - n))/b)] ))/((2 + 4*j + 2*m - n)*(2 + 2*j + 2*m + n)*(2 + 2*m + 3*n)*Sqrt[a*x^j + b *x^n])
Time = 0.43 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1938, 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.
\(\displaystyle \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1938 |
\(\displaystyle \frac {(c x)^m x^{-m-\frac {n}{2}} \sqrt {a x^j+b x^n} \int x^{m+\frac {3 n}{2}} \left (a x^{j-n}+b\right )^{3/2}dx}{\sqrt {a x^{j-n}+b}}\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \frac {b (c x)^m x^{-m-\frac {n}{2}} \sqrt {a x^j+b x^n} \int x^{m+\frac {3 n}{2}} \left (\frac {a x^{j-n}}{b}+1\right )^{3/2}dx}{\sqrt {\frac {a x^{j-n}}{b}+1}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 b x^{n+1} (c x)^m \sqrt {a x^j+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m+\frac {3 n}{2}+1}{j-n},\frac {m+\frac {3 n}{2}+1}{j-n}+1,-\frac {a x^{j-n}}{b}\right )}{(2 m+3 n+2) \sqrt {\frac {a x^{j-n}}{b}+1}}\) |
Input:
Int[(c*x)^m*(a*x^j + b*x^n)^(3/2),x]
Output:
(2*b*x^(1 + n)*(c*x)^m*Sqrt[a*x^j + b*x^n]*Hypergeometric2F1[-3/2, (1 + m + (3*n)/2)/(j - n), 1 + (1 + m + (3*n)/2)/(j - n), -((a*x^(j - n))/b)])/(( 2 + 2*m + 3*n)*Sqrt[1 + (a*x^(j - n))/b])
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[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])) Int[x^(m + j* p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !Inte gerQ[p] && NeQ[n, j] && PosQ[n - j]
\[\int \left (c x \right )^{m} \left (a \,x^{j}+b \,x^{n}\right )^{\frac {3}{2}}d x\]
Input:
int((c*x)^m*(a*x^j+b*x^n)^(3/2),x)
Output:
int((c*x)^m*(a*x^j+b*x^n)^(3/2),x)
Exception generated. \[ \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c*x)^m*(a*x^j+b*x^n)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx=\int \left (c x\right )^{m} \left (a x^{j} + b x^{n}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((c*x)**m*(a*x**j+b*x**n)**(3/2),x)
Output:
Integral((c*x)**m*(a*x**j + b*x**n)**(3/2), x)
\[ \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx=\int { {\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}} \left (c x\right )^{m} \,d x } \] Input:
integrate((c*x)^m*(a*x^j+b*x^n)^(3/2),x, algorithm="maxima")
Output:
integrate((a*x^j + b*x^n)^(3/2)*(c*x)^m, x)
\[ \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx=\int { {\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}} \left (c x\right )^{m} \,d x } \] Input:
integrate((c*x)^m*(a*x^j+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate((a*x^j + b*x^n)^(3/2)*(c*x)^m, x)
Timed out. \[ \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx=\int {\left (c\,x\right )}^m\,{\left (a\,x^j+b\,x^n\right )}^{3/2} \,d x \] Input:
int((c*x)^m*(a*x^j + b*x^n)^(3/2),x)
Output:
int((c*x)^m*(a*x^j + b*x^n)^(3/2), x)
\[ \int (c x)^m \left (a x^j+b x^n\right )^{3/2} \, dx=\text {too large to display} \] Input:
int((c*x)^m*(a*x^j+b*x^n)^(3/2),x)
Output:
(c**m*(2*x**(j + m)*sqrt(x**j*a + x**n*b)*a*j*x + 4*x**(j + m)*sqrt(x**j*a + x**n*b)*a*m*x + 4*x**(j + m)*sqrt(x**j*a + x**n*b)*a*n*x + 4*x**(j + m) *sqrt(x**j*a + x**n*b)*a*x + 8*x**(m + n)*sqrt(x**j*a + x**n*b)*b*j*x + 4* x**(m + n)*sqrt(x**j*a + x**n*b)*b*m*x - 2*x**(m + n)*sqrt(x**j*a + x**n*b )*b*n*x + 4*x**(m + n)*sqrt(x**j*a + x**n*b)*b*x + 9*int((x**(m + 2*n)*sqr t(x**j*a + x**n*b))/(3*x**j*a*j**2 + 8*x**j*a*j*m + 6*x**j*a*j*n + 8*x**j* a*j + 4*x**j*a*m**2 + 4*x**j*a*m*n + 8*x**j*a*m + 4*x**j*a*n + 4*x**j*a + 3*x**n*b*j**2 + 8*x**n*b*j*m + 6*x**n*b*j*n + 8*x**n*b*j + 4*x**n*b*m**2 + 4*x**n*b*m*n + 8*x**n*b*m + 4*x**n*b*n + 4*x**n*b),x)*b**2*j**4 + 24*int( (x**(m + 2*n)*sqrt(x**j*a + x**n*b))/(3*x**j*a*j**2 + 8*x**j*a*j*m + 6*x** j*a*j*n + 8*x**j*a*j + 4*x**j*a*m**2 + 4*x**j*a*m*n + 8*x**j*a*m + 4*x**j* a*n + 4*x**j*a + 3*x**n*b*j**2 + 8*x**n*b*j*m + 6*x**n*b*j*n + 8*x**n*b*j + 4*x**n*b*m**2 + 4*x**n*b*m*n + 8*x**n*b*m + 4*x**n*b*n + 4*x**n*b),x)*b* *2*j**3*m + 24*int((x**(m + 2*n)*sqrt(x**j*a + x**n*b))/(3*x**j*a*j**2 + 8 *x**j*a*j*m + 6*x**j*a*j*n + 8*x**j*a*j + 4*x**j*a*m**2 + 4*x**j*a*m*n + 8 *x**j*a*m + 4*x**j*a*n + 4*x**j*a + 3*x**n*b*j**2 + 8*x**n*b*j*m + 6*x**n* b*j*n + 8*x**n*b*j + 4*x**n*b*m**2 + 4*x**n*b*m*n + 8*x**n*b*m + 4*x**n*b* n + 4*x**n*b),x)*b**2*j**3 + 12*int((x**(m + 2*n)*sqrt(x**j*a + x**n*b))/( 3*x**j*a*j**2 + 8*x**j*a*j*m + 6*x**j*a*j*n + 8*x**j*a*j + 4*x**j*a*m**2 + 4*x**j*a*m*n + 8*x**j*a*m + 4*x**j*a*n + 4*x**j*a + 3*x**n*b*j**2 + 8*...