Integrand size = 19, antiderivative size = 267 \[ \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx=-\frac {2 c^2 (c x)^{7/2}}{3 b \sqrt {a+b x^3}}+\frac {7 c^5 \sqrt {c x} \sqrt {a+b x^3}}{6 b^2}-\frac {7 a^{2/3} c^5 \sqrt {c x} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} x}{\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{12 \sqrt [4]{3} b^2 \sqrt {\frac {\sqrt [3]{b} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\sqrt [3]{a}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:
-2/3*c^2*(c*x)^(7/2)/b/(b*x^3+a)^(1/2)+7/6*c^5*(c*x)^(1/2)*(b*x^3+a)^(1/2) /b^2-7/36*a^(2/3)*c^5*(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)*InverseJacob iAM(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)/b^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 = 10.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.28 \[ \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {c^5 \sqrt {c x} \left (7 a+3 b x^3-7 a \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {b x^3}{a}\right )\right )}{6 b^2 \sqrt {a+b x^3}} \] Input:
Integrate[(c*x)^(11/2)/(a + b*x^3)^(3/2),x]
Output:
(c^5*Sqrt[c*x]*(7*a + 3*b*x^3 - 7*a*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[ 1/6, 1/2, 7/6, -((b*x^3)/a)]))/(6*b^2*Sqrt[a + b*x^3])
Time = 0.59 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {817, 843, 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.
| \(\displaystyle \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {7 c^3 \int \frac {(c x)^{5/2}}{\sqrt {b x^3+a}}dx}{3 b}-\frac {2 c^2 (c x)^{7/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {7 c^3 \left (\frac {c^2 \sqrt {c x} \sqrt {a+b x^3}}{2 b}-\frac {a c^3 \int \frac {1}{\sqrt {c x} \sqrt {b x^3+a}}dx}{4 b}\right )}{3 b}-\frac {2 c^2 (c x)^{7/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 851 |
| \(\displaystyle \frac {7 c^3 \left (\frac {c^2 \sqrt {c x} \sqrt {a+b x^3}}{2 b}-\frac {a c^2 \int \frac {1}{\sqrt {b x^3+a}}d\sqrt {c x}}{2 b}\right )}{3 b}-\frac {2 c^2 (c x)^{7/2}}{3 b \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {7 c^3 \left (\frac {c^2 \sqrt {c x} \sqrt {a+b x^3}}{2 b}-\frac {a^{2/3} c \sqrt {c x} \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right ) \sqrt {\frac {a^{2/3} c^2-\sqrt [3]{a} \sqrt [3]{b} c^2 x+b^{2/3} c^2 x^2}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}{\left (1+\sqrt {3}\right ) \sqrt [3]{b} x c+\sqrt [3]{a} c}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} b \sqrt {a+b x^3} \sqrt {\frac {\sqrt [3]{b} c x \left (\sqrt [3]{a} c+\sqrt [3]{b} c x\right )}{\left (\sqrt [3]{a} c+\left (1+\sqrt {3}\right ) \sqrt [3]{b} c x\right )^2}}}\right )}{3 b}-\frac {2 c^2 (c x)^{7/2}}{3 b \sqrt {a+b x^3}}\) |
Input:
Int[(c*x)^(11/2)/(a + b*x^3)^(3/2),x]
Output:
(-2*c^2*(c*x)^(7/2))/(3*b*Sqrt[a + b*x^3]) + (7*c^3*((c^2*Sqrt[c*x]*Sqrt[a + b*x^3])/(2*b) - (a^(2/3)*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 + Sqr t[3])*b^(1/3)*c*x)^2]*EllipticF[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])/(4*3^(1/4 )*b*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])))/(3*b)
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^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && 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 = 2.88 (sec) , antiderivative size = 758, normalized size of antiderivative = 2.84
| method | result | size |
| elliptic | \(\frac {\sqrt {c x}\, \sqrt {c x \left (b \,x^{3}+a \right )}\, \left (\frac {2 c^{6} x a}{3 b^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b c x}}+\frac {c^{5} \sqrt {b c \,x^{4}+a c x}}{2 b^{2}}-\frac {7 a \,c^{6} \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 )}{6 b \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 )}{c x \sqrt {b \,x^{3}+a}}\) | \(758\) |
| risch | \(\text {Expression too large to display}\) | \(1431\) |
| default | \(\text {Expression too large to display}\) | \(2026\) |
Input:
int((c*x)^(11/2)/(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/c/x*(c*x)^(1/2)/(b*x^3+a)^(1/2)*(c*x*(b*x^3+a))^(1/2)*(2/3/b^2*c^6*x*a/( (x^3+a/b)*b*c*x)^(1/2)+1/2*c^5/b^2*(b*c*x^4+a*c*x)^(1/2)-7/6*a*c^6/b*(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)*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)))
\[ \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {11}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(11/2)/(b*x^3+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(b*x^3 + a)*sqrt(c*x)*c^5*x^5/(b^2*x^6 + 2*a*b*x^3 + a^2), x)
Result contains complex when optimal does not.
Time = 98.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.16 \[ \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {c^{\frac {11}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {13}{6} \\ \frac {19}{6} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {19}{6}\right )} \] Input:
integrate((c*x)**(11/2)/(b*x**3+a)**(3/2),x)
Output:
c**(11/2)*x**(13/2)*gamma(13/6)*hyper((3/2, 13/6), (19/6,), b*x**3*exp_pol ar(I*pi)/a)/(3*a**(3/2)*gamma(19/6))
\[ \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {11}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(11/2)/(b*x^3+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x)^(11/2)/(b*x^3 + a)^(3/2), x)
\[ \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {11}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((c*x)^(11/2)/(b*x^3+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x)^(11/2)/(b*x^3 + a)^(3/2), x)
Timed out. \[ \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int \frac {{\left (c\,x\right )}^{11/2}}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \] Input:
int((c*x)^(11/2)/(a + b*x^3)^(3/2),x)
Output:
int((c*x)^(11/2)/(a + b*x^3)^(3/2), x)
\[ \int \frac {(c x)^{11/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {\sqrt {c}\, c^{5} \left (14 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, a +4 \sqrt {x}\, \sqrt {b \,x^{3}+a}\, b \,x^{3}-7 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b^{2} x^{7}+2 a b \,x^{4}+a^{2} x}d x \right ) a^{3}-7 \left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{3}+a}}{b^{2} x^{7}+2 a b \,x^{4}+a^{2} x}d x \right ) a^{2} b \,x^{3}\right )}{8 b^{2} \left (b \,x^{3}+a \right )} \] Input:
int((c*x)^(11/2)/(b*x^3+a)^(3/2),x)
Output:
(sqrt(c)*c**5*(14*sqrt(x)*sqrt(a + b*x**3)*a + 4*sqrt(x)*sqrt(a + b*x**3)* b*x**3 - 7*int((sqrt(x)*sqrt(a + b*x**3))/(a**2*x + 2*a*b*x**4 + b**2*x**7 ),x)*a**3 - 7*int((sqrt(x)*sqrt(a + b*x**3))/(a**2*x + 2*a*b*x**4 + b**2*x **7),x)*a**2*b*x**3))/(8*b**2*(a + b*x**3))