\(\int \frac {x^3 (c+d x)^n}{a+b x^4} \, dx\) [70]

Optimal result

Integrand size = 20, antiderivative size = 349 \[ \int \frac {x^3 (c+d x)^n}{a+b x^4} \, dx=-\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c-\sqrt [4]{-a} d\right ) (1+n)}-\frac {(c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} \left (\sqrt [4]{b} c+\sqrt [4]{-a} d\right ) (1+n)} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.79 \[ \int \frac {x^3 (c+d x)^n}{a+b x^4} \, dx=\frac {(c+d x)^{1+n} \left (-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/4} (1+n)} \] Input:

Integrate[(x^3*(c + d*x)^n)/(a + b*x^4),x]
 

Output:

((c + d*x)^(1 + n)*(-(Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x 
))/(b^(1/4)*c - (-a)^(1/4)*d)]/(b^(1/4)*c - (-a)^(1/4)*d)) - Hypergeometri 
c2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)]/(b 
^(1/4)*c - I*(-a)^(1/4)*d) - Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*( 
c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]/(b^(1/4)*c + I*(-a)^(1/4)*d) - Hyp 
ergeometric2F1[1, 1 + n, 2 + n, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4 
)*d)]/(b^(1/4)*c + (-a)^(1/4)*d)))/(4*b^(3/4)*(1 + n))
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7276, 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.

\(\displaystyle \int \frac {x^3 (c+d x)^n}{a+b x^4} \, dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^3*(c + d*x)^n)/(a + b*x^4),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F]

\[\int \frac {x^{3} \left (d x +c \right )^{n}}{b \,x^{4}+a}d x\]

Input:

int(x^3*(d*x+c)^n/(b*x^4+a),x)
 

Output:

int(x^3*(d*x+c)^n/(b*x^4+a),x)
 

Fricas [F]

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

integrate(x^3*(d*x+c)^n/(b*x^4+a),x, algorithm="fricas")
 

Output:

integral((d*x + c)^n*x^3/(b*x^4 + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 (c+d x)^n}{a+b x^4} \, dx=\text {Timed out} \] Input:

integrate(x**3*(d*x+c)**n/(b*x**4+a),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^3*(d*x+c)^n/(b*x^4+a),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^n*x^3/(b*x^4 + a), x)
 

Giac [F]

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

integrate(x^3*(d*x+c)^n/(b*x^4+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^n*x^3/(b*x^4 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (c+d x)^n}{a+b x^4} \, dx=\int \frac {x^3\,{\left (c+d\,x\right )}^n}{b\,x^4+a} \,d x \] Input:

int((x^3*(c + d*x)^n)/(a + b*x^4),x)
 

Output:

int((x^3*(c + d*x)^n)/(a + b*x^4), x)
 

Reduce [F]

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

int(x^3*(d*x+c)^n/(b*x^4+a),x)
 

Output:

((c + d*x)**n - int((c + d*x)**n/(a*c + a*d*x + b*c*x**4 + b*d*x**5),x)*a* 
d*n + int(((c + d*x)**n*x**3)/(a*c + a*d*x + b*c*x**4 + b*d*x**5),x)*b*c*n 
)/(b*n)