\(\int \frac {x^9 (c+d x^4)}{(a+b x^4)^{7/4}} \, dx\) [135]

Optimal result

Integrand size = 22, antiderivative size = 152 \[ \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {a (b c-a d) x^2}{3 b^3 \left (a+b x^4\right )^{3/4}}+\frac {(7 b c-13 a d) x^2 \sqrt [4]{a+b x^4}}{21 b^3}+\frac {d x^6 \sqrt [4]{a+b x^4}}{7 b^2}-\frac {4 a^{3/2} (7 b c-10 a d) \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{21 b^{7/2} \left (a+b x^4\right )^{3/4}} \] Output:

1/3*a*(-a*d+b*c)*x^2/b^3/(b*x^4+a)^(3/4)+1/21*(-13*a*d+7*b*c)*x^2*(b*x^4+a 
)^(1/4)/b^3+1/7*d*x^6*(b*x^4+a)^(1/4)/b^2-4/21*a^(3/2)*(-10*a*d+7*b*c)*(1+ 
b*x^4/a)^(3/4)*InverseJacobiAM(1/2*arctan(b^(1/2)*x^2/a^(1/2)),2^(1/2))/b^ 
(7/2)/(b*x^4+a)^(3/4)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.68 \[ \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {x^2 \left (-20 a^2 d+2 a b \left (7 c-5 d x^4\right )+b^2 x^4 \left (7 c+3 d x^4\right )+2 a (-7 b c+10 a d) \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {3}{2},-\frac {b x^4}{a}\right )\right )}{21 b^3 \left (a+b x^4\right )^{3/4}} \] Input:

Integrate[(x^9*(c + d*x^4))/(a + b*x^4)^(7/4),x]
 

Output:

(x^2*(-20*a^2*d + 2*a*b*(7*c - 5*d*x^4) + b^2*x^4*(7*c + 3*d*x^4) + 2*a*(- 
7*b*c + 10*a*d)*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -(( 
b*x^4)/a)]))/(21*b^3*(a + b*x^4)^(3/4))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {957, 807, 262, 262, 231, 229}

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.

\(\displaystyle \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx\)

\(\Big \downarrow \) 957

\(\displaystyle \frac {x^{10} (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}-\frac {(7 b c-10 a d) \int \frac {x^9}{\left (b x^4+a\right )^{3/4}}dx}{3 a b}\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {x^{10} (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}-\frac {(7 b c-10 a d) \int \frac {x^8}{\left (b x^4+a\right )^{3/4}}dx^2}{6 a b}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {x^{10} (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}-\frac {(7 b c-10 a d) \left (\frac {2 x^6 \sqrt [4]{a+b x^4}}{7 b}-\frac {6 a \int \frac {x^4}{\left (b x^4+a\right )^{3/4}}dx^2}{7 b}\right )}{6 a b}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {x^{10} (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}-\frac {(7 b c-10 a d) \left (\frac {2 x^6 \sqrt [4]{a+b x^4}}{7 b}-\frac {6 a \left (\frac {2 x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac {2 a \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx^2}{3 b}\right )}{7 b}\right )}{6 a b}\)

\(\Big \downarrow \) 231

\(\displaystyle \frac {x^{10} (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}-\frac {(7 b c-10 a d) \left (\frac {2 x^6 \sqrt [4]{a+b x^4}}{7 b}-\frac {6 a \left (\frac {2 x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac {2 a \left (\frac {b x^4}{a}+1\right )^{3/4} \int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{3/4}}dx^2}{3 b \left (a+b x^4\right )^{3/4}}\right )}{7 b}\right )}{6 a b}\)

\(\Big \downarrow \) 229

\(\displaystyle \frac {x^{10} (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}-\frac {(7 b c-10 a d) \left (\frac {2 x^6 \sqrt [4]{a+b x^4}}{7 b}-\frac {6 a \left (\frac {2 x^2 \sqrt [4]{a+b x^4}}{3 b}-\frac {4 a^{3/2} \left (\frac {b x^4}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{3 b^{3/2} \left (a+b x^4\right )^{3/4}}\right )}{7 b}\right )}{6 a b}\)

Input:

Int[(x^9*(c + d*x^4))/(a + b*x^4)^(7/4),x]
 

Output:

((b*c - a*d)*x^10)/(3*a*b*(a + b*x^4)^(3/4)) - ((7*b*c - 10*a*d)*((2*x^6*( 
a + b*x^4)^(1/4))/(7*b) - (6*a*((2*x^2*(a + b*x^4)^(1/4))/(3*b) - (4*a^(3/ 
2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3 
*b^(3/2)*(a + b*x^4)^(3/4))))/(7*b)))/(6*a*b)
 

Defintions of rubi rules used

rule 229

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2/(a^(3/4)*Rt[b/a, 2]) 
)*EllipticF[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 231

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(3/4)/( 
a + b*x^2)^(3/4)   Int[1/(1 + b*(x^2/a))^(3/4), x], x] /; FreeQ[{a, b}, x] 
&& PosQ[a]
 

rule 262

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

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

rule 957

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a 
*b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* 
(p + 1))   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, 
 m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

Maple [F]

\[\int \frac {x^{9} \left (d \,x^{4}+c \right )}{\left (b \,x^{4}+a \right )^{\frac {7}{4}}}d x\]

Input:

int(x^9*(d*x^4+c)/(b*x^4+a)^(7/4),x)
 

Output:

int(x^9*(d*x^4+c)/(b*x^4+a)^(7/4),x)
 

Fricas [F]

\[ \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{9}}{{\left (b x^{4} + a\right )}^{\frac {7}{4}}} \,d x } \] Input:

integrate(x^9*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="fricas")
 

Output:

integral((d*x^13 + c*x^9)*(b*x^4 + a)^(1/4)/(b^2*x^8 + 2*a*b*x^4 + a^2), x 
)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 17.59 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.39 \[ \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {c x^{10} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{10 a^{\frac {7}{4}}} + \frac {d x^{14} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{14 a^{\frac {7}{4}}} \] Input:

integrate(x**9*(d*x**4+c)/(b*x**4+a)**(7/4),x)
 

Output:

c*x**10*hyper((7/4, 5/2), (7/2,), b*x**4*exp_polar(I*pi)/a)/(10*a**(7/4)) 
+ d*x**14*hyper((7/4, 7/2), (9/2,), b*x**4*exp_polar(I*pi)/a)/(14*a**(7/4) 
)
 

Maxima [F]

\[ \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{9}}{{\left (b x^{4} + a\right )}^{\frac {7}{4}}} \,d x } \] Input:

integrate(x^9*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="maxima")
 

Output:

integrate((d*x^4 + c)*x^9/(b*x^4 + a)^(7/4), x)
 

Giac [F]

\[ \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{9}}{{\left (b x^{4} + a\right )}^{\frac {7}{4}}} \,d x } \] Input:

integrate(x^9*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((d*x^4 + c)*x^9/(b*x^4 + a)^(7/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\int \frac {x^9\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{7/4}} \,d x \] Input:

int((x^9*(c + d*x^4))/(a + b*x^4)^(7/4),x)
 

Output:

int((x^9*(c + d*x^4))/(a + b*x^4)^(7/4), x)
 

Reduce [F]

\[ \int \frac {x^9 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\left (\int \frac {x^{13}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a +\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{4}}d x \right ) d +\left (\int \frac {x^{9}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a +\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{4}}d x \right ) c \] Input:

int(x^9*(d*x^4+c)/(b*x^4+a)^(7/4),x)
 

Output:

int(x**13/((a + b*x**4)**(3/4)*a + (a + b*x**4)**(3/4)*b*x**4),x)*d + int( 
x**9/((a + b*x**4)**(3/4)*a + (a + b*x**4)**(3/4)*b*x**4),x)*c