Integrand size = 19, antiderivative size = 550 \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {2}{3 a c \sqrt {c x} \sqrt {a+b x^3}}-\frac {8 \sqrt {a+b x^3}}{3 a^2 c \sqrt {c x}}+\frac {8 \left (1+\sqrt {3}\right ) \sqrt [3]{b} \sqrt {c x} \sqrt {a+b x^3}}{3 a^2 c^2 \left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )}-\frac {8 \sqrt [3]{b} \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} E\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3^{3/4} a^{5/3} c^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {4 \left (1-\sqrt {3}\right ) \sqrt [3]{b} \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{3 \sqrt [4]{3} a^{5/3} c^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
2/3/a/c/(c*x)^(1/2)/(b*x^3+a)^(1/2)-8/3*(b*x^3+a)^(1/2)/a^2/c/(c*x)^(1/2)+ 8/3*(1+3^(1/2))*b^(1/3)*(c*x)^(1/2)*(b*x^3+a)^(1/2)/a^2/c^2/(a^(1/3)+(1+3^ (1/2))*b^(1/3)*x)-8/3*b^(1/3)*(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)*Elli pticE((1-(a^(1/3)+(1-3^(1/2))*b^(1/3)*x)^2/(a^(1/3)+(1+3^(1/2))*b^(1/3)*x) ^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))*3^(1/4)/a^(5/3)/c^2/(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)-4/9*( 1-3^(1/2))*b^(1/3)*(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^(5/3)/c^2/(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 = 57, normalized size of antiderivative = 0.10 \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 x \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {3}{2},\frac {5}{6},-\frac {b x^3}{a}\right )}{a (c x)^{3/2} \sqrt {a+b x^3}} \] Input:
Integrate[1/((c*x)^(3/2)*(a + b*x^3)^(3/2)),x]
Output:
(-2*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[-1/6, 3/2, 5/6, -((b*x^3)/a)]) /(a*(c*x)^(3/2)*Sqrt[a + b*x^3])
Time = 1.11 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {819, 847, 851, 837, 25, 766, 2420}
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.
| \(\displaystyle \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {4 \int \frac {1}{(c x)^{3/2} \sqrt {b x^3+a}}dx}{3 a}+\frac {2}{3 a c \sqrt {c x} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {4 \left (\frac {2 b \int \frac {(c x)^{3/2}}{\sqrt {b x^3+a}}dx}{a c^3}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{3 a}+\frac {2}{3 a c \sqrt {c x} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {4 \left (\frac {4 b \int \frac {c^2 x^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{3 a}+\frac {2}{3 a c \sqrt {c x} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {4 \left (\frac {4 b \left (-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\int -\frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{3 a}+\frac {2}{3 a c \sqrt {c x} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \left (\frac {4 b \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) a^{2/3} c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}\right )}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{3 a}+\frac {2}{3 a c \sqrt {c x} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {4 \left (\frac {4 b \left (\frac {\int \frac {2 b^{2/3} x^2 c^2+\left (1-\sqrt {3}\right ) a^{2/3} c^2}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{3 a}+\frac {2}{3 a c \sqrt {c x} \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {4 \left (\frac {4 b \left (\frac {\frac {\left (1+\sqrt {3}\right ) c^3 \sqrt {c x} \sqrt {a+b x^3}}{\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x}-\frac {\sqrt [4]{3} \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} E\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}}{2 b^{2/3}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b^{2/3} \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{a c^4}-\frac {2 \sqrt {a+b x^3}}{a c \sqrt {c x}}\right )}{3 a}+\frac {2}{3 a c \sqrt {c x} \sqrt {a+b x^3}}\) |
Input:
Int[1/((c*x)^(3/2)*(a + b*x^3)^(3/2)),x]
Output:
2/(3*a*c*Sqrt[c*x]*Sqrt[a + b*x^3]) + (4*((-2*Sqrt[a + b*x^3])/(a*c*Sqrt[c *x]) + (4*b*((((1 + Sqrt[3])*c^3*Sqrt[c*x]*Sqrt[a + b*x^3])/(a^(1/3)*c + ( 1 + Sqrt[3])*b^(1/3)*c*x) - (3^(1/4)*a^(1/3)*c*Sqrt[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]*EllipticE[ArcCos[(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])/(Sqrt[(b^(1/3)*c*x*(a^(1/3)*c + b^(1/3)*c*x))/(a^(1/3)*c + (1 + Sq rt[3])*b^(1/3)*c*x)^2]*Sqrt[a + b*x^3]))/(2*b^(2/3)) - ((1 - Sqrt[3])*a^(1 /3)*c*Sqrt[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]*El lipticF[ArcCos[(a^(1/3)*c + (1 - Sqrt[3])*b^(1/3)*c*x)/(a^(1/3)*c + (1 + S qrt[3])*b^(1/3)*c*x)], (2 + Sqrt[3])/4])/(4*3^(1/4)*b^(2/3)*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])))/(a*c^4)))/(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[(x_)^4/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(Sqrt[3] - 1)*(s^2/(2*r^2)) Int[1/Sqrt[ a + b*x^6], x], x] - Simp[1/(2*r^2) Int[((Sqrt[3] - 1)*s^2 - 2*r^2*x^4)/S qrt[a + b*x^6], x], 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*(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]
Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*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 *r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) )*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 - Sqrt[3])*d, 0]
Result contains complex when optimal does not.
Time = 3.57 (sec) , antiderivative size = 1150, normalized size of antiderivative = 2.09
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1150\) |
| risch | \(\text {Expression too large to display}\) | \(2199\) |
| default | \(\text {Expression too large to display}\) | \(2881\) |
Input:
int(1/(c*x)^(3/2)/(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*(b*c*x^3+a*c)/a^2/c^ 2/(x*(b*c*x^3+a*c))^(1/2)-2/3*b/c*x^3/a^2/((x^3+a/b)*b*c*x)^(1/2)+8/3/a^2/ c*b*(x*(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/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)*((( -1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))/b*(-a*b^2)^(1/3)+1/b ^2*(-a*b^2)^(2/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))* b/(-a*b^2)^(1/3)*EllipticF(((-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))+(1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1 /3))*EllipticE(((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*...
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.14 \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {2 \, {\left (4 \, {\left (b x^{4} + a x\right )} \sqrt {a c} {\rm weierstrassZeta}\left (0, -\frac {4 \, b}{a}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, b}{a}, \frac {1}{x}\right )\right ) + \sqrt {b x^{3} + a} \sqrt {c x} a\right )}}{3 \, {\left (a^{2} b c^{2} x^{4} + a^{3} c^{2} x\right )}} \] Input:
integrate(1/(c*x)^(3/2)/(b*x^3+a)^(3/2),x, algorithm="fricas")
Output:
2/3*(4*(b*x^4 + a*x)*sqrt(a*c)*weierstrassZeta(0, -4*b/a, weierstrassPInve rse(0, -4*b/a, 1/x)) + sqrt(b*x^3 + a)*sqrt(c*x)*a)/(a^2*b*c^2*x^4 + a^3*c ^2*x)
Result contains complex when optimal does not.
Time = 1.77 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.09 \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {3}{2} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {5}{6}\right )} \] Input:
integrate(1/(c*x)**(3/2)/(b*x**3+a)**(3/2),x)
Output:
gamma(-1/6)*hyper((-1/6, 3/2), (5/6,), b*x**3*exp_polar(I*pi)/a)/(3*a**(3/ 2)*c**(3/2)*sqrt(x)*gamma(5/6))
\[ \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(3/2)/(b*x^3+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^3 + a)^(3/2)*(c*x)^(3/2)), x)
\[ \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(c*x)^(3/2)/(b*x^3+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((b*x^3 + a)^(3/2)*(c*x)^(3/2)), x)
Timed out. \[ \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x\right )}^{3/2}\,{\left (b\,x^3+a\right )}^{3/2}} \,d x \] Input:
int(1/((c*x)^(3/2)*(a + b*x^3)^(3/2)),x)
Output:
int(1/((c*x)^(3/2)*(a + b*x^3)^(3/2)), x)
\[ \int \frac {1}{(c x)^{3/2} \left (a+b x^3\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b^{2} x^{8}+2 a b \,x^{5}+a^{2} x^{2}}d x \right )}{c^{2}} \] Input:
int(1/(c*x)^(3/2)/(b*x^3+a)^(3/2),x)
Output:
(sqrt(c)*int((sqrt(x)*sqrt(a + b*x**3))/(a**2*x**2 + 2*a*b*x**5 + b**2*x** 8),x))/c**2