Integrand size = 15, antiderivative size = 69 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {a \sqrt {a+c x^4}}{6 x^6}-\frac {2 c \sqrt {a+c x^4}}{3 x^2}+\frac {1}{2} c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \] Output:
-1/6*a*(c*x^4+a)^(1/2)/x^6-2/3*c*(c*x^4+a)^(1/2)/x^2+1/2*c^(3/2)*arctanh(c ^(1/2)*x^2/(c*x^4+a)^(1/2))
Time = 0.17 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx=\frac {\left (-a-4 c x^4\right ) \sqrt {a+c x^4}}{6 x^6}+\frac {1}{2} c^{3/2} \log \left (\sqrt {c} x^2+\sqrt {a+c x^4}\right ) \] Input:
Integrate[(a + c*x^4)^(3/2)/x^7,x]
Output:
((-a - 4*c*x^4)*Sqrt[a + c*x^4])/(6*x^6) + (c^(3/2)*Log[Sqrt[c]*x^2 + Sqrt [a + c*x^4]])/2
Time = 0.32 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {807, 247, 247, 224, 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.
\(\displaystyle \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {\left (c x^4+a\right )^{3/2}}{x^8}dx^2\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {1}{2} \left (c \int \frac {\sqrt {c x^4+a}}{x^4}dx^2-\frac {\left (a+c x^4\right )^{3/2}}{3 x^6}\right )\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {1}{2} \left (c \left (c \int \frac {1}{\sqrt {c x^4+a}}dx^2-\frac {\sqrt {a+c x^4}}{x^2}\right )-\frac {\left (a+c x^4\right )^{3/2}}{3 x^6}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} \left (c \left (c \int \frac {1}{1-c x^4}d\frac {x^2}{\sqrt {c x^4+a}}-\frac {\sqrt {a+c x^4}}{x^2}\right )-\frac {\left (a+c x^4\right )^{3/2}}{3 x^6}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (c \left (\sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )-\frac {\sqrt {a+c x^4}}{x^2}\right )-\frac {\left (a+c x^4\right )^{3/2}}{3 x^6}\right )\) |
Input:
Int[(a + c*x^4)^(3/2)/x^7,x]
Output:
(-1/3*(a + c*x^4)^(3/2)/x^6 + c*(-(Sqrt[a + c*x^4]/x^2) + Sqrt[c]*ArcTanh[ (Sqrt[c]*x^2)/Sqrt[a + c*x^4]]))/2
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])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[(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]
Time = 0.86 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+a}\, \left (4 c \,x^{4}+a \right )}{6 x^{6}}+\frac {c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2}\) | \(47\) |
pseudoelliptic | \(\frac {3 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {c \,x^{4}+a}}{x^{2} \sqrt {c}}\right ) x^{6}-\sqrt {c \,x^{4}+a}\, \left (4 c \,x^{4}+a \right )}{6 x^{6}}\) | \(51\) |
default | \(\frac {c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2}-\frac {a \sqrt {c \,x^{4}+a}}{6 x^{6}}-\frac {2 c \sqrt {c \,x^{4}+a}}{3 x^{2}}\) | \(55\) |
elliptic | \(\frac {c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2}-\frac {a \sqrt {c \,x^{4}+a}}{6 x^{6}}-\frac {2 c \sqrt {c \,x^{4}+a}}{3 x^{2}}\) | \(55\) |
Input:
int((c*x^4+a)^(3/2)/x^7,x,method=_RETURNVERBOSE)
Output:
-1/6*(c*x^4+a)^(1/2)*(4*c*x^4+a)/x^6+1/2*c^(3/2)*ln(c^(1/2)*x^2+(c*x^4+a)^ (1/2))
Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx=\left [\frac {3 \, c^{\frac {3}{2}} x^{6} \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) - 2 \, {\left (4 \, c x^{4} + a\right )} \sqrt {c x^{4} + a}}{12 \, x^{6}}, -\frac {3 \, \sqrt {-c} c x^{6} \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-c}}{c x^{2}}\right ) + {\left (4 \, c x^{4} + a\right )} \sqrt {c x^{4} + a}}{6 \, x^{6}}\right ] \] Input:
integrate((c*x^4+a)^(3/2)/x^7,x, algorithm="fricas")
Output:
[1/12*(3*c^(3/2)*x^6*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) - 2 *(4*c*x^4 + a)*sqrt(c*x^4 + a))/x^6, -1/6*(3*sqrt(-c)*c*x^6*arctan(sqrt(c* x^4 + a)*sqrt(-c)/(c*x^2)) + (4*c*x^4 + a)*sqrt(c*x^4 + a))/x^6]
Time = 1.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.16 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx=- \frac {a \sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{6 x^{4}} - \frac {2 c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{3} - \frac {c^{\frac {3}{2}} \log {\left (\frac {a}{c x^{4}} \right )}}{4} + \frac {c^{\frac {3}{2}} \log {\left (\sqrt {\frac {a}{c x^{4}} + 1} + 1 \right )}}{2} \] Input:
integrate((c*x**4+a)**(3/2)/x**7,x)
Output:
-a*sqrt(c)*sqrt(a/(c*x**4) + 1)/(6*x**4) - 2*c**(3/2)*sqrt(a/(c*x**4) + 1) /3 - c**(3/2)*log(a/(c*x**4))/4 + c**(3/2)*log(sqrt(a/(c*x**4) + 1) + 1)/2
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {1}{4} \, c^{\frac {3}{2}} \log \left (-\frac {\sqrt {c} - \frac {\sqrt {c x^{4} + a}}{x^{2}}}{\sqrt {c} + \frac {\sqrt {c x^{4} + a}}{x^{2}}}\right ) - \frac {\sqrt {c x^{4} + a} c}{2 \, x^{2}} - \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, x^{6}} \] Input:
integrate((c*x^4+a)^(3/2)/x^7,x, algorithm="maxima")
Output:
-1/4*c^(3/2)*log(-(sqrt(c) - sqrt(c*x^4 + a)/x^2)/(sqrt(c) + sqrt(c*x^4 + a)/x^2)) - 1/2*sqrt(c*x^4 + a)*c/x^2 - 1/6*(c*x^4 + a)^(3/2)/x^6
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (53) = 106\).
Time = 0.14 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx=-\frac {1}{4} \, c^{\frac {3}{2}} \log \left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} a c^{\frac {3}{2}} - 3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} a^{2} c^{\frac {3}{2}} + 2 \, a^{3} c^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{3}} \] Input:
integrate((c*x^4+a)^(3/2)/x^7,x, algorithm="giac")
Output:
-1/4*c^(3/2)*log((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2) + 2/3*(3*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^4*a*c^(3/2) - 3*(sqrt(c)*x^2 - sqrt(c*x^4 + a))^2*a^2*c ^(3/2) + 2*a^3*c^(3/2))/((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2 - a)^3
Timed out. \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx=\int \frac {{\left (c\,x^4+a\right )}^{3/2}}{x^7} \,d x \] Input:
int((a + c*x^4)^(3/2)/x^7,x)
Output:
int((a + c*x^4)^(3/2)/x^7, x)
Time = 0.21 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.90 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^7} \, dx=\frac {3 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a c \,x^{6}+12 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{2} x^{10}-3 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{2} x^{2}-16 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a c \,x^{6}-16 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, c^{2} x^{10}+9 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{2} x^{8}+12 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{3} x^{12}-a^{3}-9 a^{2} c \,x^{4}-24 a \,c^{2} x^{8}-16 c^{3} x^{12}}{6 x^{6} \left (\sqrt {c \,x^{4}+a}\, a +4 \sqrt {c \,x^{4}+a}\, c \,x^{4}+3 \sqrt {c}\, a \,x^{2}+4 \sqrt {c}\, c \,x^{6}\right )} \] Input:
int((c*x^4+a)^(3/2)/x^7,x)
Output:
(3*sqrt(c)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) + sqrt(c)*x**2)/sqrt(a)) *a*c*x**6 + 12*sqrt(c)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) + sqrt(c)*x* *2)/sqrt(a))*c**2*x**10 - 3*sqrt(c)*sqrt(a + c*x**4)*a**2*x**2 - 16*sqrt(c )*sqrt(a + c*x**4)*a*c*x**6 - 16*sqrt(c)*sqrt(a + c*x**4)*c**2*x**10 + 9*l og((sqrt(a + c*x**4) + sqrt(c)*x**2)/sqrt(a))*a*c**2*x**8 + 12*log((sqrt(a + c*x**4) + sqrt(c)*x**2)/sqrt(a))*c**3*x**12 - a**3 - 9*a**2*c*x**4 - 24 *a*c**2*x**8 - 16*c**3*x**12)/(6*x**6*(sqrt(a + c*x**4)*a + 4*sqrt(a + c*x **4)*c*x**4 + 3*sqrt(c)*a*x**2 + 4*sqrt(c)*c*x**6))