Integrand size = 22, antiderivative size = 122 \[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=-\frac {c \sqrt [4]{a+b x^4}}{6 a x^6}+\frac {(5 b c-6 a d) \sqrt [4]{a+b x^4}}{12 a^2 x^2}+\frac {\sqrt {b} (5 b c-6 a d) \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{12 a^{3/2} \left (a+b x^4\right )^{3/4}} \] Output:
-1/6*c*(b*x^4+a)^(1/4)/a/x^6+1/12*(-6*a*d+5*b*c)*(b*x^4+a)^(1/4)/a^2/x^2+1 /12*b^(1/2)*(-6*a*d+5*b*c)*(1+b*x^4/a)^(3/4)*InverseJacobiAM(1/2*arctan(b^ (1/2)*x^2/a^(1/2)),2^(1/2))/a^(3/2)/(b*x^4+a)^(3/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\frac {-2 c \left (a+b x^4\right )+(5 b c-6 a d) x^4 \left (1+\frac {b x^4}{a}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {1}{2},-\frac {b x^4}{a}\right )}{12 a x^6 \left (a+b x^4\right )^{3/4}} \] Input:
Integrate[(c + d*x^4)/(x^7*(a + b*x^4)^(3/4)),x]
Output:
(-2*c*(a + b*x^4) + (5*b*c - 6*a*d)*x^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometr ic2F1[-1/2, 3/4, 1/2, -((b*x^4)/a)])/(12*a*x^6*(a + b*x^4)^(3/4))
Time = 0.39 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {955, 807, 264, 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.
\(\displaystyle \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle -\frac {(5 b c-6 a d) \int \frac {1}{x^3 \left (b x^4+a\right )^{3/4}}dx}{6 a}-\frac {c \sqrt [4]{a+b x^4}}{6 a x^6}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {(5 b c-6 a d) \int \frac {1}{x^4 \left (b x^4+a\right )^{3/4}}dx^2}{12 a}-\frac {c \sqrt [4]{a+b x^4}}{6 a x^6}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {(5 b c-6 a d) \left (-\frac {b \int \frac {1}{\left (b x^4+a\right )^{3/4}}dx^2}{2 a}-\frac {\sqrt [4]{a+b x^4}}{a x^2}\right )}{12 a}-\frac {c \sqrt [4]{a+b x^4}}{6 a x^6}\) |
\(\Big \downarrow \) 231 |
\(\displaystyle -\frac {(5 b c-6 a d) \left (-\frac {b \left (\frac {b x^4}{a}+1\right )^{3/4} \int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{3/4}}dx^2}{2 a \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{a x^2}\right )}{12 a}-\frac {c \sqrt [4]{a+b x^4}}{6 a x^6}\) |
\(\Big \downarrow \) 229 |
\(\displaystyle -\frac {(5 b c-6 a d) \left (-\frac {\sqrt {b} \left (\frac {b x^4}{a}+1\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ),2\right )}{\sqrt {a} \left (a+b x^4\right )^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{a x^2}\right )}{12 a}-\frac {c \sqrt [4]{a+b x^4}}{6 a x^6}\) |
Input:
Int[(c + d*x^4)/(x^7*(a + b*x^4)^(3/4)),x]
Output:
-1/6*(c*(a + b*x^4)^(1/4))/(a*x^6) - ((5*b*c - 6*a*d)*(-((a + b*x^4)^(1/4) /(a*x^2)) - (Sqrt[b]*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/ Sqrt[a]]/2, 2])/(Sqrt[a]*(a + b*x^4)^(3/4))))/(12*a)
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]
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]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
\[\int \frac {d \,x^{4}+c}{x^{7} \left (b \,x^{4}+a \right )^{\frac {3}{4}}}d x\]
Input:
int((d*x^4+c)/x^7/(b*x^4+a)^(3/4),x)
Output:
int((d*x^4+c)/x^7/(b*x^4+a)^(3/4),x)
\[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{7}} \,d x } \] Input:
integrate((d*x^4+c)/x^7/(b*x^4+a)^(3/4),x, algorithm="fricas")
Output:
integral((b*x^4 + a)^(1/4)*(d*x^4 + c)/(b*x^11 + a*x^7), x)
Result contains complex when optimal does not.
Time = 3.80 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.54 \[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=- \frac {c {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac {3}{4}} x^{6}} - \frac {d {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} x^{2}} \] Input:
integrate((d*x**4+c)/x**7/(b*x**4+a)**(3/4),x)
Output:
-c*hyper((-3/2, 3/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(3/4)*x**6) - d*hyper((-1/2, 3/4), (1/2,), b*x**4*exp_polar(I*pi)/a)/(2*a**(3/4)*x**2 )
\[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{7}} \,d x } \] Input:
integrate((d*x^4+c)/x^7/(b*x^4+a)^(3/4),x, algorithm="maxima")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(3/4)*x^7), x)
\[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int { \frac {d x^{4} + c}{{\left (b x^{4} + a\right )}^{\frac {3}{4}} x^{7}} \,d x } \] Input:
integrate((d*x^4+c)/x^7/(b*x^4+a)^(3/4),x, algorithm="giac")
Output:
integrate((d*x^4 + c)/((b*x^4 + a)^(3/4)*x^7), x)
Timed out. \[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\int \frac {d\,x^4+c}{x^7\,{\left (b\,x^4+a\right )}^{3/4}} \,d x \] Input:
int((c + d*x^4)/(x^7*(a + b*x^4)^(3/4)),x)
Output:
int((c + d*x^4)/(x^7*(a + b*x^4)^(3/4)), x)
\[ \int \frac {c+d x^4}{x^7 \left (a+b x^4\right )^{3/4}} \, dx=\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{7}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {3}{4}} x^{3}}d x \right ) d \] Input:
int((d*x^4+c)/x^7/(b*x^4+a)^(3/4),x)
Output:
int(1/((a + b*x**4)**(3/4)*x**7),x)*c + int(1/((a + b*x**4)**(3/4)*x**3),x )*d