Integrand size = 21, antiderivative size = 168 \[ \int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=-\frac {(c+d x)^n \left (a-b x^2\right )^{5/2}}{5 b}+\frac {(c+d x)^n \left (a-b x^2\right )^{5/2} \operatorname {AppellF1}\left (n,-\frac {5}{2},-\frac {5}{2},1+n,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{5 b \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{5/2} \left (1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{5/2}} \] Output:
-1/5*(d*x+c)^n*(-b*x^2+a)^(5/2)/b+1/5*(d*x+c)^n*(-b*x^2+a)^(5/2)*AppellF1( n,-5/2,-5/2,1+n,(d*x+c)/(c-a^(1/2)*d/b^(1/2)),(d*x+c)/(c+a^(1/2)*d/b^(1/2) ))/b/(1-(d*x+c)/(c-a^(1/2)*d/b^(1/2)))^(5/2)/(1-(d*x+c)/(c+a^(1/2)*d/b^(1/ 2)))^(5/2)
\[ \int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx \] Input:
Integrate[x*(c + d*x)^n*(a - b*x^2)^(3/2),x]
Output:
Integrate[x*(c + d*x)^n*(a - b*x^2)^(3/2), x]
Time = 0.34 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.77, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {624, 514, 150}
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.
\(\displaystyle \int x \left (a-b x^2\right )^{3/2} (c+d x)^n \, dx\) |
\(\Big \downarrow \) 624 |
\(\displaystyle \frac {\int (c+d x)^{n+1} \left (a-b x^2\right )^{3/2}dx}{d}-\frac {c \int (c+d x)^n \left (a-b x^2\right )^{3/2}dx}{d}\) |
\(\Big \downarrow \) 514 |
\(\displaystyle \frac {\left (a-b x^2\right )^{3/2} \int (c+d x)^{n+1} \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2} \left (1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2}d(c+d x)}{d^2 \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2} \left (1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}\right )^{3/2}}-\frac {c \left (a-b x^2\right )^{3/2} \int (c+d x)^n \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2} \left (1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2}d(c+d x)}{d^2 \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2} \left (1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}\right )^{3/2}}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {\left (a-b x^2\right )^{3/2} (c+d x)^{n+2} \operatorname {AppellF1}\left (n+2,-\frac {3}{2},-\frac {3}{2},n+3,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d^2 (n+2) \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2} \left (1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}\right )^{3/2}}-\frac {c \left (a-b x^2\right )^{3/2} (c+d x)^{n+1} \operatorname {AppellF1}\left (n+1,-\frac {3}{2},-\frac {3}{2},n+2,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d^2 (n+1) \left (1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}\right )^{3/2} \left (1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}\right )^{3/2}}\) |
Input:
Int[x*(c + d*x)^n*(a - b*x^2)^(3/2),x]
Output:
-((c*(c + d*x)^(1 + n)*(a - b*x^2)^(3/2)*AppellF1[1 + n, -3/2, -3/2, 2 + n , (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b]) ])/(d^2*(1 + n)*(1 - (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b]))^(3/2)*(1 - (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b]))^(3/2))) + ((c + d*x)^(2 + n)*(a - b*x^2)^( 3/2)*AppellF1[2 + n, -3/2, -3/2, 3 + n, (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b] ), (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b])])/(d^2*(2 + n)*(1 - (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b]))^(3/2)*(1 - (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b]))^(3/2 ))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( c + d*x)/(c + d*q))^p) Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[1/d Int[x^(m - 1)*(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] - Si mp[c/d Int[x^(m - 1)*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && IGtQ[m, 0]
\[\int x \left (d x +c \right )^{n} \left (-b \,x^{2}+a \right )^{\frac {3}{2}}d x\]
Input:
int(x*(d*x+c)^n*(-b*x^2+a)^(3/2),x)
Output:
int(x*(d*x+c)^n*(-b*x^2+a)^(3/2),x)
\[ \int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{n} x \,d x } \] Input:
integrate(x*(d*x+c)^n*(-b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
integral(-(b*x^3 - a*x)*sqrt(-b*x^2 + a)*(d*x + c)^n, x)
\[ \int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int x \left (a - b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{n}\, dx \] Input:
integrate(x*(d*x+c)**n*(-b*x**2+a)**(3/2),x)
Output:
Integral(x*(a - b*x**2)**(3/2)*(c + d*x)**n, x)
\[ \int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{n} x \,d x } \] Input:
integrate(x*(d*x+c)^n*(-b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((-b*x^2 + a)^(3/2)*(d*x + c)^n*x, x)
\[ \int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int { {\left (-b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{n} x \,d x } \] Input:
integrate(x*(d*x+c)^n*(-b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((-b*x^2 + a)^(3/2)*(d*x + c)^n*x, x)
Timed out. \[ \int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=\int x\,{\left (a-b\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^n \,d x \] Input:
int(x*(a - b*x^2)^(3/2)*(c + d*x)^n,x)
Output:
int(x*(a - b*x^2)^(3/2)*(c + d*x)^n, x)
\[ \int x (c+d x)^n \left (a-b x^2\right )^{3/2} \, dx=-\left (\int \left (d x +c \right )^{n} \sqrt {-b \,x^{2}+a}\, x^{3}d x \right ) b +\left (\int \left (d x +c \right )^{n} \sqrt {-b \,x^{2}+a}\, x d x \right ) a \] Input:
int(x*(d*x+c)^n*(-b*x^2+a)^(3/2),x)
Output:
- int((c + d*x)**n*sqrt(a - b*x**2)*x**3,x)*b + int((c + d*x)**n*sqrt(a - b*x**2)*x,x)*a