Integrand size = 15, antiderivative size = 125 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx=\frac {b^2 x^2}{4 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{6 x^6}-\frac {b \left (a+b x^4\right )^{3/4}}{4 a x^2}-\frac {b^{3/2} \sqrt [4]{1+\frac {b x^4}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{4 \sqrt {a} \sqrt [4]{a+b x^4}} \] Output:
1/4*b^2*x^2/a/(b*x^4+a)^(1/4)-1/6*(b*x^4+a)^(3/4)/x^6-1/4*b*(b*x^4+a)^(3/4 )/a/x^2-1/4*b^(3/2)*(1+b*x^4/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x^2 /a^(1/2))),2^(1/2))/a^(1/2)/(b*x^4+a)^(1/4)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.41 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx=-\frac {\left (a+b x^4\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{4},-\frac {1}{2},-\frac {b x^4}{a}\right )}{6 x^6 \left (1+\frac {b x^4}{a}\right )^{3/4}} \] Input:
Integrate[(a + b*x^4)^(3/4)/x^7,x]
Output:
-1/6*((a + b*x^4)^(3/4)*Hypergeometric2F1[-3/2, -3/4, -1/2, -((b*x^4)/a)]) /(x^6*(1 + (b*x^4)/a)^(3/4))
Time = 0.37 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {807, 247, 264, 227, 225, 212}
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.
\(\displaystyle \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^4+a\right )^{3/4}}{x^8}dx^2\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} b \int \frac {1}{x^4 \sqrt [4]{b x^4+a}}dx^2-\frac {\left (a+b x^4\right )^{3/4}}{3 x^6}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} b \left (\frac {b \int \frac {1}{\sqrt [4]{b x^4+a}}dx^2}{2 a}-\frac {\left (a+b x^4\right )^{3/4}}{a x^2}\right )-\frac {\left (a+b x^4\right )^{3/4}}{3 x^6}\right )\) |
\(\Big \downarrow \) 227 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} b \left (\frac {b \sqrt [4]{\frac {b x^4}{a}+1} \int \frac {1}{\sqrt [4]{\frac {b x^4}{a}+1}}dx^2}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{a x^2}\right )-\frac {\left (a+b x^4\right )^{3/4}}{3 x^6}\right )\) |
\(\Big \downarrow \) 225 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} b \left (\frac {b \sqrt [4]{\frac {b x^4}{a}+1} \left (\frac {2 x^2}{\sqrt [4]{\frac {b x^4}{a}+1}}-\int \frac {1}{\left (\frac {b x^4}{a}+1\right )^{5/4}}dx^2\right )}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{a x^2}\right )-\frac {\left (a+b x^4\right )^{3/4}}{3 x^6}\right )\) |
\(\Big \downarrow \) 212 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} b \left (\frac {b \sqrt [4]{\frac {b x^4}{a}+1} \left (\frac {2 x^2}{\sqrt [4]{\frac {b x^4}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{2 a \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{a x^2}\right )-\frac {\left (a+b x^4\right )^{3/4}}{3 x^6}\right )\) |
Input:
Int[(a + b*x^4)^(3/4)/x^7,x]
Output:
(-1/3*(a + b*x^4)^(3/4)/x^6 + (b*(-((a + b*x^4)^(3/4)/(a*x^2)) + (b*(1 + ( b*x^4)/a)^(1/4)*((2*x^2)/(1 + (b*x^4)/a)^(1/4) - (2*Sqrt[a]*EllipticE[ArcT an[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/Sqrt[b]))/(2*a*(a + b*x^4)^(1/4))))/2)/2
Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) )*EllipticE[(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)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) , x] - Simp[a Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ a, 0] && PosQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( a + b*x^2)^(1/4) Int[1/(1 + b*(x^2/a))^(1/4), 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/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1))) Int[ (c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && LtQ[m, -1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
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]
\[\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x^{7}}d x\]
Input:
int((b*x^4+a)^(3/4)/x^7,x)
Output:
int((b*x^4+a)^(3/4)/x^7,x)
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{7}} \,d x } \] Input:
integrate((b*x^4+a)^(3/4)/x^7,x, algorithm="fricas")
Output:
integral((b*x^4 + a)^(3/4)/x^7, x)
Result contains complex when optimal does not.
Time = 0.63 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.27 \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx=- \frac {a^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {3}{4} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{6 x^{6}} \] Input:
integrate((b*x**4+a)**(3/4)/x**7,x)
Output:
-a**(3/4)*hyper((-3/2, -3/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*x**6)
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{7}} \,d x } \] Input:
integrate((b*x^4+a)^(3/4)/x^7,x, algorithm="maxima")
Output:
integrate((b*x^4 + a)^(3/4)/x^7, x)
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx=\int { \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{7}} \,d x } \] Input:
integrate((b*x^4+a)^(3/4)/x^7,x, algorithm="giac")
Output:
integrate((b*x^4 + a)^(3/4)/x^7, x)
Timed out. \[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx=\int \frac {{\left (b\,x^4+a\right )}^{3/4}}{x^7} \,d x \] Input:
int((a + b*x^4)^(3/4)/x^7,x)
Output:
int((a + b*x^4)^(3/4)/x^7, x)
\[ \int \frac {\left (a+b x^4\right )^{3/4}}{x^7} \, dx=\frac {-\left (b \,x^{4}+a \right )^{\frac {3}{4}}-3 \left (\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{b \,x^{11}+a \,x^{7}}d x \right ) a \,x^{6}}{3 x^{6}} \] Input:
int((b*x^4+a)^(3/4)/x^7,x)
Output:
( - (a + b*x**4)**(3/4) - 3*int((a + b*x**4)**(3/4)/(a*x**7 + b*x**11),x)* a*x**6)/(3*x**6)