Integrand size = 19, antiderivative size = 296 \[ \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {2}{3 a c (c x)^{11/2} \sqrt {a+b x^3}}-\frac {28 \sqrt {a+b x^3}}{33 a^2 c (c x)^{11/2}}+\frac {224 b \sqrt {a+b x^3}}{165 a^3 c^4 (c x)^{5/2}}+\frac {224 b^2 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{165 \sqrt [4]{3} a^{10/3} c^7 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2/3/a/c/(c*x)^(11/2)/(b*x^3+a)^(1/2)-28/33*(b*x^3+a)^(1/2)/a^2/c/(c*x)^(11 /2)+224/165*b*(b*x^3+a)^(1/2)/a^3/c^4/(c*x)^(5/2)+224/495*b^2*(c*x)^(1/2)* (a^(1/3)+b^(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(a^(1/3)+(1+3 ^(1/2))*b^(1/3)*x)^2)^(1/2)*InverseJacobiAM(arccos((a^(1/3)+(1-3^(1/2))*b^ (1/3)*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x)),1/4*6^(1/2)+1/4*2^(1/2))*3^(3/4) /a^(10/3)/c^7/(b^(1/3)*x*(a^(1/3)+b^(1/3)*x)/(a^(1/3)+(1+3^(1/2))*b^(1/3)* x)^2)^(1/2)/(b*x^3+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.20 \[ \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 x \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {11}{6},\frac {3}{2},-\frac {5}{6},-\frac {b x^3}{a}\right )}{11 a (c x)^{13/2} \sqrt {a+b x^3}} \] Input:
Integrate[1/((c*x)^(13/2)*(a + b*x^3)^(3/2)),x]
Output:
(-2*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-11/6, 3/2, -5/6, -((b*x^3)/a) ])/(11*a*(c*x)^(13/2)*Sqrt[a + b*x^3])
Time = 0.63 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {819, 847, 847, 851, 766}
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.
| \(\displaystyle \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {14 \int \frac {1}{(c x)^{13/2} \sqrt {b x^3+a}}dx}{3 a}+\frac {2}{3 a c (c x)^{11/2} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {14 \left (-\frac {8 b \int \frac {1}{(c x)^{7/2} \sqrt {b x^3+a}}dx}{11 a c^3}-\frac {2 \sqrt {a+b x^3}}{11 a c (c x)^{11/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{11/2} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {14 \left (-\frac {8 b \left (-\frac {2 b \int \frac {1}{\sqrt {c x} \sqrt {b x^3+a}}dx}{5 a c^3}-\frac {2 \sqrt {a+b x^3}}{5 a c (c x)^{5/2}}\right )}{11 a c^3}-\frac {2 \sqrt {a+b x^3}}{11 a c (c x)^{11/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{11/2} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {14 \left (-\frac {8 b \left (-\frac {4 b \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{5 a c^4}-\frac {2 \sqrt {a+b x^3}}{5 a c (c x)^{5/2}}\right )}{11 a c^3}-\frac {2 \sqrt {a+b x^3}}{11 a c (c x)^{11/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{11/2} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {14 \left (-\frac {8 b \left (-\frac {2 b \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{5 \sqrt [4]{3} a^{4/3} c^5 \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}-\frac {2 \sqrt {a+b x^3}}{5 a c (c x)^{5/2}}\right )}{11 a c^3}-\frac {2 \sqrt {a+b x^3}}{11 a c (c x)^{11/2}}\right )}{3 a}+\frac {2}{3 a c (c x)^{11/2} \sqrt {a+b x^3}}\) |
Input:
Int[1/((c*x)^(13/2)*(a + b*x^3)^(3/2)),x]
Output:
2/(3*a*c*(c*x)^(11/2)*Sqrt[a + b*x^3]) + (14*((-2*Sqrt[a + b*x^3])/(11*a*c *(c*x)^(11/2)) - (8*b*((-2*Sqrt[a + b*x^3])/(5*a*c*(c*x)^(5/2)) - (2*b*Sqr t[c*x]*(a^(1/3)*c + b^(1/3)*c*x)*Sqrt[(a^(2/3)*c^2 - a^(1/3)*b^(1/3)*c^2*x + b^(2/3)*c^2*x^2)/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)^2]*EllipticF[A rcCos[(a^(1/3)*c + (1 - Sqrt[3])*b^(1/3)*c*x)/(a^(1/3)*c + (1 + Sqrt[3])*b ^(1/3)*c*x)], (2 + Sqrt[3])/4])/(5*3^(1/4)*a^(4/3)*c^5*Sqrt[(b^(1/3)*c*x*( a^(1/3)*c + b^(1/3)*c*x))/(a^(1/3)*c + (1 + Sqrt[3])*b^(1/3)*c*x)^2]*Sqrt[ a + b*x^3])))/(11*a*c^3)))/(3*a)
Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ (s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x ]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Result contains complex when optimal does not.
Time = 5.02 (sec) , antiderivative size = 784, normalized size of antiderivative = 2.65
| method | result | size |
| elliptic | \(\frac {\sqrt {c x \left (b \,x^{3}+a \right )}\, \left (-\frac {2 \sqrt {b c \,x^{4}+a c x}}{11 a^{2} c^{7} x^{6}}+\frac {38 b \sqrt {b c \,x^{4}+a c x}}{55 a^{3} c^{7} x^{3}}+\frac {2 b^{2} x}{3 c^{6} a^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b c x}}+\frac {448 b^{3} \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}^{2} \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \sqrt {\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}} \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{b \left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) x}{\left (-\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}}, \sqrt {\frac {\left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}{\left (\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{165 a^{3} c^{6} \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {b c x \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{3}+a}}\) | \(784\) |
| risch | \(\text {Expression too large to display}\) | \(1445\) |
| default | \(\text {Expression too large to display}\) | \(2224\) |
Input:
int(1/(c*x)^(13/2)/(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
(c*x*(b*x^3+a))^(1/2)/(c*x)^(1/2)/(b*x^3+a)^(1/2)*(-2/11/a^2/c^7*(b*c*x^4+ a*c*x)^(1/2)/x^6+38/55*b/a^3/c^7*(b*c*x^4+a*c*x)^(1/2)/x^3+2/3*b^2/c^6*x/a ^3/((x^3+a/b)*b*c*x)^(1/2)+448/165*b^3/a^3/c^6*(1/2/b*(-a*b^2)^(1/3)-1/2*I *3^(1/2)/b*(-a*b^2)^(1/3))*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2 )^(1/3))*x/(-1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*( -a*b^2)^(1/3)))^(1/2)*(x-1/b*(-a*b^2)^(1/3))^2*(1/b*(-a*b^2)^(1/3)*(x+1/2/ b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a*b^2)^(1/3)-1/ 2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)*(1/b*(-a*b^2)^ (1/3)*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-1/2/b*(-a* b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2)/( -3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(-a*b^2)^(1/3)/(b*c* x*(x-1/b*(-a*b^2)^(1/3))*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^ (1/3))*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*Elli pticF(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*x/(-1/2/b*(- a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(x-1/b*(-a*b^2)^(1/3)))^(1/2) ,((3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*(1/2/b*(-a*b^2)^(1 /3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b* (-a*b^2)^(1/3))/(3/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/ 2)))
Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.34 \[ \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 \, {\left (224 \, {\left (b^{3} x^{9} + a b^{2} x^{6}\right )} \sqrt {a c} {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right ) - {\left (112 \, a b^{2} x^{6} + 42 \, a^{2} b x^{3} - 15 \, a^{3}\right )} \sqrt {b x^{3} + a} \sqrt {c x}\right )}}{165 \, {\left (a^{4} b c^{7} x^{9} + a^{5} c^{7} x^{6}\right )}} \] Input:
integrate(1/(c*x)^(13/2)/(b*x^3+a)^(3/2),x, algorithm="fricas")
Output:
-2/165*(224*(b^3*x^9 + a*b^2*x^6)*sqrt(a*c)*weierstrassPInverse(0, -4*b/a, 1/x) - (112*a*b^2*x^6 + 42*a^2*b*x^3 - 15*a^3)*sqrt(b*x^3 + a)*sqrt(c*x)) /(a^4*b*c^7*x^9 + a^5*c^7*x^6)
Timed out. \[ \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(c*x)**(13/2)/(b*x**3+a)**(3/2),x)
Output:
Timed out
\[ \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(13/2)/(b*x^3+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^3 + a)^(3/2)*(c*x)^(13/2)), x)
\[ \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(13/2)/(b*x^3+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((b*x^3 + a)^(3/2)*(c*x)^(13/2)), x)
Timed out. \[ \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{13/2}\,{\left (b\,x^3+a\right )}^{3/2}} \,d x \] Input:
int(1/((c*x)^(13/2)*(a + b*x^3)^(3/2)),x)
Output:
int(1/((c*x)^(13/2)*(a + b*x^3)^(3/2)), x)
\[ \int \frac {1}{(c x)^{13/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b^{2} x^{13}+2 a b \,x^{10}+a^{2} x^{7}}d x \right )}{c^{7}} \] Input:
int(1/(c*x)^(13/2)/(b*x^3+a)^(3/2),x)
Output:
(sqrt(c)*int((sqrt(x)*sqrt(a + b*x**3))/(a**2*x**7 + 2*a*b*x**10 + b**2*x* *13),x))/c**7