\(\int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx\) [214]

Optimal result

Integrand size = 20, antiderivative size = 239 \[ \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx=\frac {\left (b c^2-a d^2\right ) (c+d x)^{1+n}}{b^2 d^3 (1+n)}-\frac {2 c (c+d x)^{2+n}}{b d^3 (2+n)}+\frac {(c+d x)^{3+n}}{b d^3 (3+n)}+\frac {a^2 (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 b^2 \left (\sqrt {-a} \sqrt {b} c+a d\right ) (1+n)}+\frac {a (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 b^2 \left (\frac {\sqrt {b} c}{\sqrt {-a}}+d\right ) (1+n)} \] Output:

(-a*d^2+b*c^2)*(d*x+c)^(1+n)/b^2/d^3/(1+n)-2*c*(d*x+c)^(2+n)/b/d^3/(2+n)+( 
d*x+c)^(3+n)/b/d^3/(3+n)+1/2*a^2*(d*x+c)^(1+n)*hypergeom([1, 1+n],[2+n],b^ 
(1/2)*(d*x+c)/(b^(1/2)*c-(-a)^(1/2)*d))/b^2/((-a)^(1/2)*b^(1/2)*c+a*d)/(1+ 
n)+1/2*a*(d*x+c)^(1+n)*hypergeom([1, 1+n],[2+n],b^(1/2)*(d*x+c)/(b^(1/2)*c 
+(-a)^(1/2)*d))/b^2/(b^(1/2)*c/(-a)^(1/2)+d)/(1+n)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.91 \[ \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx=\frac {(c+d x)^{1+n} \left (\frac {2 \left (b c^2-a d^2\right )}{d^3 (1+n)}-\frac {4 b c (c+d x)}{d^3 (2+n)}+\frac {2 b (c+d x)^2}{d^3 (3+n)}+\frac {(-a)^{3/2} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{\left (\sqrt {b} c-\sqrt {-a} d\right ) (1+n)}+\frac {\sqrt {-a} a \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{\left (\sqrt {b} c+\sqrt {-a} d\right ) (1+n)}\right )}{2 b^2} \] Input:

Integrate[(x^4*(c + d*x)^n)/(a + b*x^2),x]
 

Output:

((c + d*x)^(1 + n)*((2*(b*c^2 - a*d^2))/(d^3*(1 + n)) - (4*b*c*(c + d*x))/ 
(d^3*(2 + n)) + (2*b*(c + d*x)^2)/(d^3*(3 + n)) + ((-a)^(3/2)*Hypergeometr 
ic2F1[1, 1 + n, 2 + n, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c - Sqrt[-a]*d)])/((Sq 
rt[b]*c - Sqrt[-a]*d)*(1 + n)) + (Sqrt[-a]*a*Hypergeometric2F1[1, 1 + n, 2 
 + n, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[-a]*d)])/((Sqrt[b]*c + Sqrt[-a 
]*d)*(1 + n))))/(2*b^2)
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {604, 25, 2160, 2009}

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.

\(\displaystyle \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx\)

\(\Big \downarrow \) 604

\(\displaystyle \frac {\int -\frac {(c+d x)^n \left (2 b c (n+3) x^3 d^3+2 a c (n+3) x d^3+\left (b c^2+a d^2\right ) (n+3) x^2 d^2+a c^2 (n+3) d^2\right )}{b x^2+a}dx}{b d^4 (n+3)}+\frac {(c+d x)^{n+3}}{b d^3 (n+3)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(c+d x)^{n+3}}{b d^3 (n+3)}-\frac {\int \frac {(c+d x)^n \left (2 b c (n+3) x^3 d^3+2 a c (n+3) x d^3+\left (b c^2+a d^2\right ) (n+3) x^2 d^2+a c^2 (n+3) d^2\right )}{b x^2+a}dx}{b d^4 (n+3)}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {(c+d x)^{n+3}}{b d^3 (n+3)}-\frac {\int \left (-\frac {d^2 \left (b c^2-a d^2\right ) (n+3) (c+d x)^n}{b}+\frac {\left (-\frac {a^2 n d^4}{b}-\frac {3 a^2 d^4}{b}\right ) (c+d x)^n}{b x^2+a}+2 c d^2 (n+3) (c+d x)^{n+1}\right )dx}{b d^4 (n+3)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(c+d x)^{n+3}}{b d^3 (n+3)}-\frac {-\frac {d (n+3) \left (b c^2-a d^2\right ) (c+d x)^{n+1}}{b (n+1)}-\frac {(-a)^{3/2} d^4 (n+3) (c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}\right )}{2 b (n+1) \left (\sqrt {b} c-\sqrt {-a} d\right )}+\frac {(-a)^{3/2} d^4 (n+3) (c+d x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}\right )}{2 b (n+1) \left (\sqrt {-a} d+\sqrt {b} c\right )}+\frac {2 c d (n+3) (c+d x)^{n+2}}{n+2}}{b d^4 (n+3)}\)

Input:

Int[(x^4*(c + d*x)^n)/(a + b*x^2),x]
 

Output:

(c + d*x)^(3 + n)/(b*d^3*(3 + n)) - (-((d*(b*c^2 - a*d^2)*(3 + n)*(c + d*x 
)^(1 + n))/(b*(1 + n))) + (2*c*d*(3 + n)*(c + d*x)^(2 + n))/(2 + n) - ((-a 
)^(3/2)*d^4*(3 + n)*(c + d*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, ( 
Sqrt[b]*(c + d*x))/(Sqrt[b]*c - Sqrt[-a]*d)])/(2*b*(Sqrt[b]*c - Sqrt[-a]*d 
)*(1 + n)) + ((-a)^(3/2)*d^4*(3 + n)*(c + d*x)^(1 + n)*Hypergeometric2F1[1 
, 1 + n, 2 + n, (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[-a]*d)])/(2*b*(Sqrt[ 
b]*c + Sqrt[-a]*d)*(1 + n)))/(b*d^4*(3 + n))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 604

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2160

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, 
 d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

Maple [F]

\[\int \frac {x^{4} \left (d x +c \right )^{n}}{b \,x^{2}+a}d x\]

Input:

int(x^4*(d*x+c)^n/(b*x^2+a),x)
 

Output:

int(x^4*(d*x+c)^n/(b*x^2+a),x)
 

Fricas [F]

\[ \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x^{4}}{b x^{2} + a} \,d x } \] Input:

integrate(x^4*(d*x+c)^n/(b*x^2+a),x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx=\int \frac {x^{4} \left (c + d x\right )^{n}}{a + b x^{2}}\, dx \] Input:

integrate(x**4*(d*x+c)**n/(b*x**2+a),x)
 

Output:

Integral(x**4*(c + d*x)**n/(a + b*x**2), x)
 

Maxima [F]

\[ \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x^{4}}{b x^{2} + a} \,d x } \] Input:

integrate(x^4*(d*x+c)^n/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^n*x^4/(b*x^2 + a), x)
 

Giac [F]

\[ \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{n} x^{4}}{b x^{2} + a} \,d x } \] Input:

integrate(x^4*(d*x+c)^n/(b*x^2+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^n*x^4/(b*x^2 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx=\int \frac {x^4\,{\left (c+d\,x\right )}^n}{b\,x^2+a} \,d x \] Input:

int((x^4*(c + d*x)^n)/(a + b*x^2),x)
 

Output:

int((x^4*(c + d*x)^n)/(a + b*x^2), x)
 

Reduce [F]

\[ \int \frac {x^4 (c+d x)^n}{a+b x^2} \, dx=\frac {-\left (d x +c \right )^{n} a c \,d^{2} n^{2}-5 \left (d x +c \right )^{n} a c \,d^{2} n -6 \left (d x +c \right )^{n} a c \,d^{2}-\left (d x +c \right )^{n} a \,d^{3} n^{2} x -5 \left (d x +c \right )^{n} a \,d^{3} n x -6 \left (d x +c \right )^{n} a \,d^{3} x +2 \left (d x +c \right )^{n} b \,c^{3}-2 \left (d x +c \right )^{n} b \,c^{2} d n x +\left (d x +c \right )^{n} b c \,d^{2} n^{2} x^{2}+\left (d x +c \right )^{n} b c \,d^{2} n \,x^{2}+\left (d x +c \right )^{n} b \,d^{3} n^{2} x^{3}+3 \left (d x +c \right )^{n} b \,d^{3} n \,x^{3}+2 \left (d x +c \right )^{n} b \,d^{3} x^{3}+\left (\int \frac {\left (d x +c \right )^{n}}{b \,x^{2}+a}d x \right ) a^{2} d^{3} n^{3}+6 \left (\int \frac {\left (d x +c \right )^{n}}{b \,x^{2}+a}d x \right ) a^{2} d^{3} n^{2}+11 \left (\int \frac {\left (d x +c \right )^{n}}{b \,x^{2}+a}d x \right ) a^{2} d^{3} n +6 \left (\int \frac {\left (d x +c \right )^{n}}{b \,x^{2}+a}d x \right ) a^{2} d^{3}}{b^{2} d^{3} \left (n^{3}+6 n^{2}+11 n +6\right )} \] Input:

int(x^4*(d*x+c)^n/(b*x^2+a),x)
 

Output:

( - (c + d*x)**n*a*c*d**2*n**2 - 5*(c + d*x)**n*a*c*d**2*n - 6*(c + d*x)** 
n*a*c*d**2 - (c + d*x)**n*a*d**3*n**2*x - 5*(c + d*x)**n*a*d**3*n*x - 6*(c 
 + d*x)**n*a*d**3*x + 2*(c + d*x)**n*b*c**3 - 2*(c + d*x)**n*b*c**2*d*n*x 
+ (c + d*x)**n*b*c*d**2*n**2*x**2 + (c + d*x)**n*b*c*d**2*n*x**2 + (c + d* 
x)**n*b*d**3*n**2*x**3 + 3*(c + d*x)**n*b*d**3*n*x**3 + 2*(c + d*x)**n*b*d 
**3*x**3 + int((c + d*x)**n/(a + b*x**2),x)*a**2*d**3*n**3 + 6*int((c + d* 
x)**n/(a + b*x**2),x)*a**2*d**3*n**2 + 11*int((c + d*x)**n/(a + b*x**2),x) 
*a**2*d**3*n + 6*int((c + d*x)**n/(a + b*x**2),x)*a**2*d**3)/(b**2*d**3*(n 
**3 + 6*n**2 + 11*n + 6))