\(\int \frac {x (c+d x)^n}{(a-b x^2)^{3/2}} \, dx\) [1816]

Optimal result

Integrand size = 21, antiderivative size = 163 \[ \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx=\frac {(c+d x)^n}{b \sqrt {a-b x^2}}-\frac {(c+d x)^n \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \operatorname {AppellF1}\left (n,\frac {1}{2},\frac {1}{2},1+n,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{b \sqrt {a-b x^2}} \] Output:

(d*x+c)^n/b/(-b*x^2+a)^(1/2)-(d*x+c)^n*(1-(d*x+c)/(c-a^(1/2)*d/b^(1/2)))^( 
1/2)*(1-(d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*AppellF1(n,1/2,1/2,1+n,(d*x+c 
)/(c-a^(1/2)*d/b^(1/2)),(d*x+c)/(c+a^(1/2)*d/b^(1/2)))/b/(-b*x^2+a)^(1/2)
 

Mathematica [F]

\[ \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx \] 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]
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(365\) vs. \(2(163)=326\).

Time = 0.75 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.24, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {593, 719, 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.

\(\displaystyle \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 593

\(\displaystyle \frac {(c-d x) (c+d x)^{n+1}}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {d \int \frac {(c+d x)^n (c n-d (n+1) x)}{\sqrt {a-b x^2}}dx}{b c^2-a d^2}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {(c-d x) (c+d x)^{n+1}}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {d \left (c (2 n+1) \int \frac {(c+d x)^n}{\sqrt {a-b x^2}}dx-(n+1) \int \frac {(c+d x)^{n+1}}{\sqrt {a-b x^2}}dx\right )}{b c^2-a d^2}\)

\(\Big \downarrow \) 514

\(\displaystyle \frac {(c-d x) (c+d x)^{n+1}}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {d \left (\frac {c (2 n+1) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}} \int \frac {(c+d x)^n}{\sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}}}d(c+d x)}{d \sqrt {a-b x^2}}-\frac {(n+1) \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}} \int \frac {(c+d x)^{n+1}}{\sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}}}d(c+d x)}{d \sqrt {a-b x^2}}\right )}{b c^2-a d^2}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {(c-d x) (c+d x)^{n+1}}{\sqrt {a-b x^2} \left (b c^2-a d^2\right )}-\frac {d \left (\frac {c (2 n+1) (c+d x)^{n+1} \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}} \operatorname {AppellF1}\left (n+1,\frac {1}{2},\frac {1}{2},n+2,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d (n+1) \sqrt {a-b x^2}}-\frac {(n+1) (c+d x)^{n+2} \sqrt {1-\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {1-\frac {c+d x}{\frac {\sqrt {a} d}{\sqrt {b}}+c}} \operatorname {AppellF1}\left (n+2,\frac {1}{2},\frac {1}{2},n+3,\frac {c+d x}{c-\frac {\sqrt {a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}\right )}{d (n+2) \sqrt {a-b x^2}}\right )}{b c^2-a d^2}\)

Input:

Int[(x*(c + d*x)^n)/(a - b*x^2)^(3/2),x]
 

Output:

((c - d*x)*(c + d*x)^(1 + n))/((b*c^2 - a*d^2)*Sqrt[a - b*x^2]) - (d*((c*( 
1 + 2*n)*(c + d*x)^(1 + n)*Sqrt[1 - (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b])]*S 
qrt[1 - (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b])]*AppellF1[1 + n, 1/2, 1/2, 2 + 
 n, (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b 
])])/(d*(1 + n)*Sqrt[a - b*x^2]) - ((1 + n)*(c + d*x)^(2 + n)*Sqrt[1 - (c 
+ d*x)/(c - (Sqrt[a]*d)/Sqrt[b])]*Sqrt[1 - (c + d*x)/(c + (Sqrt[a]*d)/Sqrt 
[b])]*AppellF1[2 + n, 1/2, 1/2, 3 + n, (c + d*x)/(c - (Sqrt[a]*d)/Sqrt[b]) 
, (c + d*x)/(c + (Sqrt[a]*d)/Sqrt[b])])/(d*(2 + n)*Sqrt[a - b*x^2])))/(b*c 
^2 - a*d^2)
 

Defintions of rubi rules used

rule 150

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])
 

rule 514

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]
 

rule 593

Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[(c + d*x)^(n + 1)*(c - d*x)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + 
a*d^2))), x] - Simp[d/(2*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a + b* 
x^2)^(p + 1)*(c*n - d*(n + 2*p + 4)*x), x], x] /; FreeQ[{a, b, c, d, n}, x] 
 && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [F]

\[\int \frac {x \left (d x +c \right )^{n}}{\left (-b \,x^{2}+a \right )^{\frac {3}{2}}}d x\]

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)
 

Fricas [F]

\[ \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x*(d*x+c)^n/(-b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(-b*x^2 + a)*(d*x + c)^n*x/(b^2*x^4 - 2*a*b*x^2 + a^2), x)
 

Sympy [F]

\[ \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x \left (c + d x\right )^{n}}{\left (a - b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x*(d*x+c)**n/(-b*x**2+a)**(3/2),x)
 

Output:

Integral(x*(c + d*x)**n/(a - b*x**2)**(3/2), x)
 

Maxima [F]

\[ \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x*(d*x+c)^n/(-b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^n*x/(-b*x^2 + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x*(d*x+c)^n/(-b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x + c)^n*x/(-b*x^2 + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx=\int \frac {x\,{\left (c+d\,x\right )}^n}{{\left (a-b\,x^2\right )}^{3/2}} \,d x \] Input:

int((x*(c + d*x)^n)/(a - b*x^2)^(3/2),x)
 

Output:

int((x*(c + d*x)^n)/(a - b*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x (c+d x)^n}{\left (a-b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x*(d*x+c)^n/(-b*x^2+a)^(3/2),x)
 

Output:

( - (sqrt(b)*c + sqrt(b)*d*x)**n + sqrt(a - b*x**2)*int((sqrt(b)*c + sqrt( 
b)*d*x)**n/(sqrt(a - b*x**2)*a*c**2*n - sqrt(a - b*x**2)*a*c**2 - sqrt(a - 
 b*x**2)*a*d**2*n*x**2 + sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b 
*c**2*n*x**2 + sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*n*x* 
*4 - sqrt(a - b*x**2)*b*d**2*x**4),x)*a*c*d*n**2 - sqrt(a - b*x**2)*int((s 
qrt(b)*c + sqrt(b)*d*x)**n/(sqrt(a - b*x**2)*a*c**2*n - sqrt(a - b*x**2)*a 
*c**2 - sqrt(a - b*x**2)*a*d**2*n*x**2 + sqrt(a - b*x**2)*a*d**2*x**2 - sq 
rt(a - b*x**2)*b*c**2*n*x**2 + sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x 
**2)*b*d**2*n*x**4 - sqrt(a - b*x**2)*b*d**2*x**4),x)*a*c*d*n - sqrt(a - b 
*x**2)*int(((sqrt(b)*c + sqrt(b)*d*x)**n*x**2)/(sqrt(a - b*x**2)*a*c**2*n 
- sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2*n*x**2 + sqrt(a - b*x* 
*2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*n*x**2 + sqrt(a - b*x**2)*b*c**2 
*x**2 + sqrt(a - b*x**2)*b*d**2*n*x**4 - sqrt(a - b*x**2)*b*d**2*x**4),x)* 
b*c*d*n**2 + sqrt(a - b*x**2)*int(((sqrt(b)*c + sqrt(b)*d*x)**n*x**2)/(sqr 
t(a - b*x**2)*a*c**2*n - sqrt(a - b*x**2)*a*c**2 - sqrt(a - b*x**2)*a*d**2 
*n*x**2 + sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x**2)*b*c**2*n*x**2 + 
sqrt(a - b*x**2)*b*c**2*x**2 + sqrt(a - b*x**2)*b*d**2*n*x**4 - sqrt(a - b 
*x**2)*b*d**2*x**4),x)*b*c*d*n - sqrt(a - b*x**2)*int(((sqrt(b)*c + sqrt(b 
)*d*x)**n*x)/(sqrt(a - b*x**2)*a*c**2*n - sqrt(a - b*x**2)*a*c**2 - sqrt(a 
 - b*x**2)*a*d**2*n*x**2 + sqrt(a - b*x**2)*a*d**2*x**2 - sqrt(a - b*x*...