Integrand size = 15, antiderivative size = 70 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx=-\frac {a \sqrt {a+c x^4}}{2 x^2}+\frac {1}{4} c x^2 \sqrt {a+c x^4}+\frac {3}{4} a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \] Output:
-1/2*a*(c*x^4+a)^(1/2)/x^2+1/4*c*x^2*(c*x^4+a)^(1/2)+3/4*a*c^(1/2)*arctanh (c^(1/2)*x^2/(c*x^4+a)^(1/2))
Time = 0.22 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx=\frac {\left (-2 a+c x^4\right ) \sqrt {a+c x^4}}{4 x^2}+\frac {3}{4} a \sqrt {c} \log \left (\sqrt {c} x^2+\sqrt {a+c x^4}\right ) \] Input:
Integrate[(a + c*x^4)^(3/2)/x^3,x]
Output:
((-2*a + c*x^4)*Sqrt[a + c*x^4])/(4*x^2) + (3*a*Sqrt[c]*Log[Sqrt[c]*x^2 + Sqrt[a + c*x^4]])/4
Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {807, 247, 211, 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.
\(\displaystyle \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{2} \int \frac {\left (c x^4+a\right )^{3/2}}{x^4}dx^2\) |
\(\Big \downarrow \) 247 |
\(\displaystyle \frac {1}{2} \left (3 c \int \sqrt {c x^4+a}dx^2-\frac {\left (a+c x^4\right )^{3/2}}{x^2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {1}{2} \left (3 c \left (\frac {1}{2} a \int \frac {1}{\sqrt {c x^4+a}}dx^2+\frac {1}{2} x^2 \sqrt {a+c x^4}\right )-\frac {\left (a+c x^4\right )^{3/2}}{x^2}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{2} \left (3 c \left (\frac {1}{2} a \int \frac {1}{1-c x^4}d\frac {x^2}{\sqrt {c x^4+a}}+\frac {1}{2} x^2 \sqrt {a+c x^4}\right )-\frac {\left (a+c x^4\right )^{3/2}}{x^2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (3 c \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c}}+\frac {1}{2} x^2 \sqrt {a+c x^4}\right )-\frac {\left (a+c x^4\right )^{3/2}}{x^2}\right )\) |
Input:
Int[(a + c*x^4)^(3/2)/x^3,x]
Output:
(-((a + c*x^4)^(3/2)/x^2) + 3*c*((x^2*Sqrt[a + c*x^4])/2 + (a*ArcTanh[(Sqr t[c]*x^2)/Sqrt[a + c*x^4]])/(2*Sqrt[c])))/2
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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 = 0.78 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.71
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+a}\, \left (-c \,x^{4}+2 a \right )}{4 x^{2}}+\frac {3 a \sqrt {c}\, \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{4}\) | \(50\) |
default | \(-\frac {a \sqrt {c \,x^{4}+a}}{2 x^{2}}+\frac {c \,x^{2} \sqrt {c \,x^{4}+a}}{4}+\frac {3 a \sqrt {c}\, \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{4}\) | \(56\) |
elliptic | \(-\frac {a \sqrt {c \,x^{4}+a}}{2 x^{2}}+\frac {c \,x^{2} \sqrt {c \,x^{4}+a}}{4}+\frac {3 a \sqrt {c}\, \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{4}\) | \(56\) |
pseudoelliptic | \(\frac {\sqrt {c \,x^{4}+a}\, c \,x^{4}+3 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {c \,x^{4}+a}}{x^{2} \sqrt {c}}\right ) a \,x^{2}-2 a \sqrt {c \,x^{4}+a}}{4 x^{2}}\) | \(59\) |
Input:
int((c*x^4+a)^(3/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/4*(c*x^4+a)^(1/2)*(-c*x^4+2*a)/x^2+3/4*a*c^(1/2)*ln(c^(1/2)*x^2+(c*x^4+ a)^(1/2))
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx=\left [\frac {3 \, a \sqrt {c} x^{2} \log \left (-2 \, c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {c} x^{2} - a\right ) + 2 \, \sqrt {c x^{4} + a} {\left (c x^{4} - 2 \, a\right )}}{8 \, x^{2}}, -\frac {3 \, a \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{4} + a} \sqrt {-c}}{c x^{2}}\right ) - \sqrt {c x^{4} + a} {\left (c x^{4} - 2 \, a\right )}}{4 \, x^{2}}\right ] \] Input:
integrate((c*x^4+a)^(3/2)/x^3,x, algorithm="fricas")
Output:
[1/8*(3*a*sqrt(c)*x^2*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) + 2*sqrt(c*x^4 + a)*(c*x^4 - 2*a))/x^2, -1/4*(3*a*sqrt(-c)*x^2*arctan(sqrt(c *x^4 + a)*sqrt(-c)/(c*x^2)) - sqrt(c*x^4 + a)*(c*x^4 - 2*a))/x^2]
Time = 1.35 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx=- \frac {a^{\frac {3}{2}}}{2 x^{2} \sqrt {1 + \frac {c x^{4}}{a}}} - \frac {\sqrt {a} c x^{2}}{4 \sqrt {1 + \frac {c x^{4}}{a}}} + \frac {3 a \sqrt {c} \operatorname {asinh}{\left (\frac {\sqrt {c} x^{2}}{\sqrt {a}} \right )}}{4} + \frac {c^{2} x^{6}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{4}}{a}}} \] Input:
integrate((c*x**4+a)**(3/2)/x**3,x)
Output:
-a**(3/2)/(2*x**2*sqrt(1 + c*x**4/a)) - sqrt(a)*c*x**2/(4*sqrt(1 + c*x**4/ a)) + 3*a*sqrt(c)*asinh(sqrt(c)*x**2/sqrt(a))/4 + c**2*x**6/(4*sqrt(a)*sqr t(1 + c*x**4/a))
Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx=-\frac {3}{8} \, a \sqrt {c} \log \left (-\frac {\sqrt {c} - \frac {\sqrt {c x^{4} + a}}{x^{2}}}{\sqrt {c} + \frac {\sqrt {c x^{4} + a}}{x^{2}}}\right ) - \frac {\sqrt {c x^{4} + a} a}{2 \, x^{2}} - \frac {\sqrt {c x^{4} + a} a c}{4 \, {\left (c - \frac {c x^{4} + a}{x^{4}}\right )} x^{2}} \] Input:
integrate((c*x^4+a)^(3/2)/x^3,x, algorithm="maxima")
Output:
-3/8*a*sqrt(c)*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)*a/x^2 - 1/4*sqrt(c*x^4 + a)*a*c/((c - (c* x^4 + a)/x^4)*x^2)
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx=\frac {1}{4} \, \sqrt {c x^{4} + a} c x^{2} - \frac {3}{8} \, a \sqrt {c} \log \left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2}\right ) + \frac {a^{2} \sqrt {c}}{{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a} \] Input:
integrate((c*x^4+a)^(3/2)/x^3,x, algorithm="giac")
Output:
1/4*sqrt(c*x^4 + a)*c*x^2 - 3/8*a*sqrt(c)*log((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2) + a^2*sqrt(c)/((sqrt(c)*x^2 - sqrt(c*x^4 + a))^2 - a)
Timed out. \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (c\,x^4+a\right )}^{3/2}}{x^3} \,d x \] Input:
int((a + c*x^4)^(3/2)/x^3,x)
Output:
int((a + c*x^4)^(3/2)/x^3, x)
Time = 0.24 (sec) , antiderivative size = 270, normalized size of antiderivative = 3.86 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^3} \, dx=\frac {12 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2} x^{2}+48 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a c \,x^{6}-33 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{2} x^{2}-56 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a c \,x^{6}+16 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, c^{2} x^{10}+36 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2} c \,x^{4}+48 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{2} x^{8}-8 a^{3}-63 a^{2} c \,x^{4}-48 a \,c^{2} x^{8}+16 c^{3} x^{12}}{16 x^{2} \left (\sqrt {c \,x^{4}+a}\, a +4 \sqrt {c \,x^{4}+a}\, c \,x^{4}+3 \sqrt {c}\, a \,x^{2}+4 \sqrt {c}\, c \,x^{6}\right )} \] Input:
int((c*x^4+a)^(3/2)/x^3,x)
Output:
(12*sqrt(c)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) + sqrt(c)*x**2)/sqrt(a) )*a**2*x**2 + 48*sqrt(c)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) + sqrt(c)* x**2)/sqrt(a))*a*c*x**6 - 33*sqrt(c)*sqrt(a + c*x**4)*a**2*x**2 - 56*sqrt( c)*sqrt(a + c*x**4)*a*c*x**6 + 16*sqrt(c)*sqrt(a + c*x**4)*c**2*x**10 + 36 *log((sqrt(a + c*x**4) + sqrt(c)*x**2)/sqrt(a))*a**2*c*x**4 + 48*log((sqrt (a + c*x**4) + sqrt(c)*x**2)/sqrt(a))*a*c**2*x**8 - 8*a**3 - 63*a**2*c*x** 4 - 48*a*c**2*x**8 + 16*c**3*x**12)/(16*x**2*(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))