\(\int \frac {x^6 (c+d x^4)}{(a+b x^4)^{7/4}} \, dx\) [130]

Optimal result

Integrand size = 22, antiderivative size = 126 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=-\frac {(b c-a d) x^3}{3 b^2 \left (a+b x^4\right )^{3/4}}+\frac {d x^3 \sqrt [4]{a+b x^4}}{4 b^2}-\frac {(4 b c-7 a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{11/4}}+\frac {(4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{11/4}} \] Output:

-1/3*(-a*d+b*c)*x^3/b^2/(b*x^4+a)^(3/4)+1/4*d*x^3*(b*x^4+a)^(1/4)/b^2-1/8* 
(-7*a*d+4*b*c)*arctan(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(11/4)+1/8*(-7*a*d+4*b* 
c)*arctanh(b^(1/4)*x/(b*x^4+a)^(1/4))/b^(11/4)
 

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.85 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {\frac {2 b^{3/4} x^3 \left (-4 b c+7 a d+3 b d x^4\right )}{\left (a+b x^4\right )^{3/4}}+3 (-4 b c+7 a d) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+3 (4 b c-7 a d) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{24 b^{11/4}} \] Input:

Integrate[(x^6*(c + d*x^4))/(a + b*x^4)^(7/4),x]
 

Output:

((2*b^(3/4)*x^3*(-4*b*c + 7*a*d + 3*b*d*x^4))/(a + b*x^4)^(3/4) + 3*(-4*b* 
c + 7*a*d)*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)] + 3*(4*b*c - 7*a*d)*ArcTa 
nh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(24*b^(11/4))
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {957, 843, 854, 827, 216, 219}

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.

\(\displaystyle \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx\)

\(\Big \downarrow \) 957

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

\(\Big \downarrow \) 843

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

\(\Big \downarrow \) 854

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {x^7 (b c-a d)}{3 a b \left (a+b x^4\right )^{3/4}}-\frac {(4 b c-7 a d) \left (\frac {x^3 \sqrt [4]{a+b x^4}}{4 b}-\frac {3 a \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}}\right )}{4 b}\right )}{3 a b}\)

Input:

Int[(x^6*(c + d*x^4))/(a + b*x^4)^(7/4),x]
 

Output:

((b*c - a*d)*x^7)/(3*a*b*(a + b*x^4)^(3/4)) - ((4*b*c - 7*a*d)*((x^3*(a + 
b*x^4)^(1/4))/(4*b) - (3*a*(-1/2*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/b^( 
3/4) + ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(3/4))))/(4*b)))/(3*a*b 
)
 

Defintions of rubi rules used

rule 216

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 843

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

rule 854

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

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 [A] (verified)

Time = 0.32 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(-\frac {\frac {21 \left (b \,x^{4}+a \right )^{\frac {3}{4}} b^{2} \left (a d -\frac {4 c b}{7}\right ) \ln \left (\frac {x \,b^{\frac {1}{4}}+\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{-x \,b^{\frac {1}{4}}+\left (b \,x^{4}+a \right )^{\frac {1}{4}}}\right )}{16}+\frac {21 \left (b \,x^{4}+a \right )^{\frac {3}{4}} b^{2} \left (a d -\frac {4 c b}{7}\right ) \arctan \left (\frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{x \,b^{\frac {1}{4}}}\right )}{8}+b^{\frac {11}{4}} \left (\left (-\frac {3 d \,x^{4}}{4}+c \right ) b -\frac {7 a d}{4}\right ) x^{3}}{3 \left (b \,x^{4}+a \right )^{\frac {3}{4}} b^{\frac {19}{4}}}\)	\(134\)

Input:

int(x^6*(d*x^4+c)/(b*x^4+a)^(7/4),x,method=_RETURNVERBOSE)
 

Output:

-1/3/(b*x^4+a)^(3/4)*(21/16*(b*x^4+a)^(3/4)*b^2*(a*d-4/7*c*b)*ln((x*b^(1/4 
)+(b*x^4+a)^(1/4))/(-x*b^(1/4)+(b*x^4+a)^(1/4)))+21/8*(b*x^4+a)^(3/4)*b^2* 
(a*d-4/7*c*b)*arctan((b*x^4+a)^(1/4)/x/b^(1/4))+b^(11/4)*((-3/4*d*x^4+c)*b 
-7/4*a*d)*x^3)/b^(19/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 697, normalized size of antiderivative = 5.53 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx =\text {Too large to display} \] Input:

integrate(x^6*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="fricas")
 

Output:

1/48*(3*(b^3*x^4 + a*b^2)*((256*b^4*c^4 - 1792*a*b^3*c^3*d + 4704*a^2*b^2* 
c^2*d^2 - 5488*a^3*b*c*d^3 + 2401*a^4*d^4)/b^11)^(1/4)*log(-(b^3*x*((256*b 
^4*c^4 - 1792*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 5488*a^3*b*c*d^3 + 2401 
*a^4*d^4)/b^11)^(1/4) + (b*x^4 + a)^(1/4)*(4*b*c - 7*a*d))/x) - 3*(b^3*x^4 
 + a*b^2)*((256*b^4*c^4 - 1792*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 5488*a 
^3*b*c*d^3 + 2401*a^4*d^4)/b^11)^(1/4)*log((b^3*x*((256*b^4*c^4 - 1792*a*b 
^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 5488*a^3*b*c*d^3 + 2401*a^4*d^4)/b^11)^( 
1/4) - (b*x^4 + a)^(1/4)*(4*b*c - 7*a*d))/x) - 3*(I*b^3*x^4 + I*a*b^2)*((2 
56*b^4*c^4 - 1792*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 5488*a^3*b*c*d^3 + 
2401*a^4*d^4)/b^11)^(1/4)*log((I*b^3*x*((256*b^4*c^4 - 1792*a*b^3*c^3*d + 
4704*a^2*b^2*c^2*d^2 - 5488*a^3*b*c*d^3 + 2401*a^4*d^4)/b^11)^(1/4) - (b*x 
^4 + a)^(1/4)*(4*b*c - 7*a*d))/x) - 3*(-I*b^3*x^4 - I*a*b^2)*((256*b^4*c^4 
 - 1792*a*b^3*c^3*d + 4704*a^2*b^2*c^2*d^2 - 5488*a^3*b*c*d^3 + 2401*a^4*d 
^4)/b^11)^(1/4)*log((-I*b^3*x*((256*b^4*c^4 - 1792*a*b^3*c^3*d + 4704*a^2* 
b^2*c^2*d^2 - 5488*a^3*b*c*d^3 + 2401*a^4*d^4)/b^11)^(1/4) - (b*x^4 + a)^( 
1/4)*(4*b*c - 7*a*d))/x) + 4*(3*b*d*x^7 - (4*b*c - 7*a*d)*x^3)*(b*x^4 + a) 
^(1/4))/(b^3*x^4 + a*b^2)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 14.54 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.63 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\frac {c x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{4}} \Gamma \left (\frac {11}{4}\right )} + \frac {d x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {7}{4}} \Gamma \left (\frac {15}{4}\right )} \] Input:

integrate(x**6*(d*x**4+c)/(b*x**4+a)**(7/4),x)
 

Output:

c*x**7*gamma(7/4)*hyper((7/4, 7/4), (11/4,), b*x**4*exp_polar(I*pi)/a)/(4* 
a**(7/4)*gamma(11/4)) + d*x**11*gamma(11/4)*hyper((7/4, 11/4), (15/4,), b* 
x**4*exp_polar(I*pi)/a)/(4*a**(7/4)*gamma(15/4))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (102) = 204\).

Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.83 \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=-\frac {1}{12} \, {\left (\frac {4 \, x^{3}}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} b} - \frac {3 \, {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {3}{4}}} - \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {3}{4}}}\right )}}{b}\right )} c + \frac {1}{48} \, d {\left (\frac {4 \, {\left (4 \, a b - \frac {7 \, {\left (b x^{4} + a\right )} a}{x^{4}}\right )}}{\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}} b^{3}}{x^{3}} - \frac {{\left (b x^{4} + a\right )}^{\frac {7}{4}} b^{2}}{x^{7}}} - \frac {21 \, {\left (\frac {2 \, a \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right )}{b^{\frac {3}{4}}} - \frac {a \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right )}{b^{\frac {3}{4}}}\right )}}{b^{2}}\right )} \] Input:

integrate(x^6*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="maxima")
 

Output:

-1/12*(4*x^3/((b*x^4 + a)^(3/4)*b) - 3*(2*arctan((b*x^4 + a)^(1/4)/(b^(1/4 
)*x))/b^(3/4) - log(-(b^(1/4) - (b*x^4 + a)^(1/4)/x)/(b^(1/4) + (b*x^4 + a 
)^(1/4)/x))/b^(3/4))/b)*c + 1/48*d*(4*(4*a*b - 7*(b*x^4 + a)*a/x^4)/((b*x^ 
4 + a)^(3/4)*b^3/x^3 - (b*x^4 + a)^(7/4)*b^2/x^7) - 21*(2*a*arctan((b*x^4 
+ a)^(1/4)/(b^(1/4)*x))/b^(3/4) - a*log(-(b^(1/4) - (b*x^4 + a)^(1/4)/x)/( 
b^(1/4) + (b*x^4 + a)^(1/4)/x))/b^(3/4))/b^2)
 

Giac [F]

\[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\int { \frac {{\left (d x^{4} + c\right )} x^{6}}{{\left (b x^{4} + a\right )}^{\frac {7}{4}}} \,d x } \] Input:

integrate(x^6*(d*x^4+c)/(b*x^4+a)^(7/4),x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)*x^6/(b*x^4 + a)^(7/4), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\int \frac {x^6\,\left (d\,x^4+c\right )}{{\left (b\,x^4+a\right )}^{7/4}} \,d x \] Input:

int((x^6*(c + d*x^4))/(a + b*x^4)^(7/4),x)
 

Output:

int((x^6*(c + d*x^4))/(a + b*x^4)^(7/4), x)
 

Reduce [F]

\[ \int \frac {x^6 \left (c+d x^4\right )}{\left (a+b x^4\right )^{7/4}} \, dx=\left (\int \frac {x^{10}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a +\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{4}}d x \right ) d +\left (\int \frac {x^{6}}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} a +\left (b \,x^{4}+a \right )^{\frac {3}{4}} b \,x^{4}}d x \right ) c \] Input:

int(x^6*(d*x^4+c)/(b*x^4+a)^(7/4),x)
 

Output:

int(x**10/((a + b*x**4)**(3/4)*a + (a + b*x**4)**(3/4)*b*x**4),x)*d + int( 
x**6/((a + b*x**4)**(3/4)*a + (a + b*x**4)**(3/4)*b*x**4),x)*c