Integrand size = 19, antiderivative size = 471 \[ \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {3}{b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 a^{11/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 b^7 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {23 \sqrt {b \sqrt [3]{x}+a x}}{7 b^2 x^{11/3}}+\frac {437 a \sqrt {b \sqrt [3]{x}+a x}}{119 b^3 x^3}-\frac {6555 a^2 \sqrt {b \sqrt [3]{x}+a x}}{1547 b^4 x^{7/3}}+\frac {24035 a^3 \sqrt {b \sqrt [3]{x}+a x}}{4641 b^5 x^{5/3}}-\frac {4807 a^4 \sqrt {b \sqrt [3]{x}+a x}}{663 b^6 x}+\frac {4807 a^5 \sqrt {b \sqrt [3]{x}+a x}}{221 b^7 \sqrt [3]{x}}+\frac {4807 a^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 b^{27/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {4807 a^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{442 b^{27/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:
3/b/x^(10/3)/(b*x^(1/3)+a*x)^(1/2)-4807/221*a^(11/2)*(b+a*x^(2/3))*x^(1/3) /b^7/(b^(1/2)+a^(1/2)*x^(1/3))/(b*x^(1/3)+a*x)^(1/2)-23/7*(b*x^(1/3)+a*x)^ (1/2)/b^2/x^(11/3)+437/119*a*(b*x^(1/3)+a*x)^(1/2)/b^3/x^3-6555/1547*a^2*( b*x^(1/3)+a*x)^(1/2)/b^4/x^(7/3)+24035/4641*a^3*(b*x^(1/3)+a*x)^(1/2)/b^5/ x^(5/3)-4807/663*a^4*(b*x^(1/3)+a*x)^(1/2)/b^6/x+4807/221*a^5*(b*x^(1/3)+a *x)^(1/2)/b^7/x^(1/3)+4807/221*a^(21/4)*(b^(1/2)+a^(1/2)*x^(1/3))*((b+a*x^ (2/3))/(b^(1/2)+a^(1/2)*x^(1/3))^2)^(1/2)*x^(1/6)*EllipticE(sin(2*arctan(a ^(1/4)*x^(1/6)/b^(1/4))),1/2*2^(1/2))/b^(27/4)/(b*x^(1/3)+a*x)^(1/2)-4807/ 442*a^(21/4)*(b^(1/2)+a^(1/2)*x^(1/3))*((b+a*x^(2/3))/(b^(1/2)+a^(1/2)*x^( 1/3))^2)^(1/2)*x^(1/6)*InverseJacobiAM(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)),1 /2*2^(1/2))/b^(27/4)/(b*x^(1/3)+a*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {21}{4},\frac {3}{2},-\frac {17}{4},-\frac {a x^{2/3}}{b}\right )}{7 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}} \] Input:
Integrate[1/(x^4*(b*x^(1/3) + a*x)^(3/2)),x]
Output:
(-2*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[-21/4, 3/2, -17/4, -((a*x^(2 /3))/b)])/(7*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])
Time = 1.28 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {1924, 1929, 1931, 1931, 1931, 1931, 1931, 1931, 1938, 266, 834, 27, 761, 1510}
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.
| \(\displaystyle \int \frac {1}{x^4 \left (a x+b \sqrt [3]{x}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 3 \int \frac {1}{x^{10/3} \left (\sqrt [3]{x} b+a x\right )^{3/2}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 1929 |
| \(\displaystyle 3 \left (\frac {23 \int \frac {1}{x^{11/3} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \int \frac {1}{x^3 \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \int \frac {1}{x^{7/3} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \int \frac {1}{x^{5/3} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \int \frac {1}{x \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \left (-\frac {3 a \int \frac {1}{\sqrt [3]{x} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \left (-\frac {3 a \left (\frac {a \int \frac {\sqrt [3]{x}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1938 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \left (-\frac {3 a \left (\frac {a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {\sqrt [6]{x}}{\sqrt {x^{2/3} a+b}}d\sqrt [3]{x}}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 834 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {b} \sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle 3 \left (\frac {23 \left (-\frac {19 a \left (-\frac {15 a \left (-\frac {11 a \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a x^{4/3}+b}}-\frac {\sqrt [6]{x} \sqrt {a x^{4/3}+b}}{\sqrt {a} x^{2/3}+\sqrt {b}}}{\sqrt {a}}\right )}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{13 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{13 b x^{7/3}}\right )}{17 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{17 b x^3}\right )}{21 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{21 b x^{11/3}}\right )}{2 b}+\frac {1}{b x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\) |
Input:
Int[1/(x^4*(b*x^(1/3) + a*x)^(3/2)),x]
Output:
3*(1/(b*x^(10/3)*Sqrt[b*x^(1/3) + a*x]) + (23*((-2*Sqrt[b*x^(1/3) + a*x])/ (21*b*x^(11/3)) - (19*a*((-2*Sqrt[b*x^(1/3) + a*x])/(17*b*x^3) - (15*a*((- 2*Sqrt[b*x^(1/3) + a*x])/(13*b*x^(7/3)) - (11*a*((-2*Sqrt[b*x^(1/3) + a*x] )/(9*b*x^(5/3)) - (7*a*((-2*Sqrt[b*x^(1/3) + a*x])/(5*b*x) - (3*a*((-2*Sqr t[b*x^(1/3) + a*x])/(b*x^(1/3)) + (2*a*Sqrt[b + a*x^(2/3)]*x^(1/6)*(-((-(( x^(1/6)*Sqrt[b + a*x^(4/3)])/(Sqrt[b] + Sqrt[a]*x^(2/3))) + (b^(1/4)*(Sqrt [b] + Sqrt[a]*x^(2/3))*Sqrt[(b + a*x^(4/3))/(Sqrt[b] + Sqrt[a]*x^(2/3))^2] *EllipticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(a^(1/4)*Sqrt[b + a* x^(4/3)]))/Sqrt[a]) + (b^(1/4)*(Sqrt[b] + Sqrt[a]*x^(2/3))*Sqrt[(b + a*x^( 4/3))/(Sqrt[b] + Sqrt[a]*x^(2/3))^2]*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/ b^(1/4)], 1/2])/(2*a^(3/4)*Sqrt[b + a*x^(4/3)])))/(b*Sqrt[b*x^(1/3) + a*x] )))/(5*b)))/(9*b)))/(13*b)))/(17*b)))/(21*b)))/(2*b))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S imp[1/q Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp [1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x ], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j ] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1 ]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] & & !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))) I nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[ m + j*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])) Int[x^(m + j* p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !Inte gerQ[p] && NeQ[n, j] && PosQ[n - j]
Time = 6.17 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(-\frac {2 \sqrt {b \,x^{\frac {1}{3}}+a x}}{7 b^{2} x^{\frac {11}{3}}}+\frac {80 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{119 b^{3} x^{3}}-\frac {1914 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 b^{4} x^{\frac {7}{3}}}+\frac {10112 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 b^{5} x^{\frac {5}{3}}}-\frac {2818 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{663 b^{6} x}+\frac {4144 \left (b +a \,x^{\frac {2}{3}}\right ) a^{5}}{221 b^{7} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}+\frac {3 x^{\frac {2}{3}} a^{6}}{b^{7} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}-\frac {4807 a^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{442 b^{7} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(333\) |
| default | \(\frac {-201894 a^{5} b \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {20}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \operatorname {EllipticE}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+100947 a^{5} b \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {20}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+201894 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {22}{3}} a^{6}+174048 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {20}{3}} a^{5} b -19228 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, x^{6} a^{4} b^{2}-39452 x^{\frac {20}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{5} b -5244 x^{\frac {14}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b^{4}+8740 x^{\frac {16}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3} b^{3}+3588 x^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{5}-2652 x^{\frac {10}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{6}}{9282 x^{7} \left (b +a \,x^{\frac {2}{3}}\right ) b^{7}}\) | \(411\) |
Input:
int(1/x^4/(b*x^(1/3)+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/7*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(11/3)+80/119*a*(b*x^(1/3)+a*x)^(1/2)/b^3 /x^3-1914/1547*a^2*(b*x^(1/3)+a*x)^(1/2)/b^4/x^(7/3)+10112/4641*a^3*(b*x^( 1/3)+a*x)^(1/2)/b^5/x^(5/3)-2818/663*a^4*(b*x^(1/3)+a*x)^(1/2)/b^6/x+4144/ 221*(b+a*x^(2/3))*a^5/b^7/(x^(1/3)*(b+a*x^(2/3)))^(1/2)+3*x^(2/3)*a^6/b^7/ ((x^(2/3)+b/a)*x^(1/3)*a)^(1/2)-4807/442*a^5/b^7*(-a*b)^(1/2)*((x^(1/3)+1/ a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2)*(-2*(x^(1/3)-1/a*(-a*b)^(1/2))*a/(-a *b)^(1/2))^(1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)/(b*x^(1/3)+a*x)^(1/2)*(-2 /a*(-a*b)^(1/2)*EllipticE(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2 ),1/2*2^(1/2))+1/a*(-a*b)^(1/2)*EllipticF(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(- a*b)^(1/2))^(1/2),1/2*2^(1/2)))
\[ \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^(1/3)+a*x)^(3/2),x, algorithm="fricas")
Output:
integral((a^4*x^3 + 3*a^2*b^2*x^(5/3) - 2*a*b^3*x - (2*a^3*b*x^2 - b^4)*x^ (1/3))*sqrt(a*x + b*x^(1/3))/(a^6*x^9 + 2*a^3*b^3*x^7 + b^6*x^5), x)
\[ \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{4} \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**4/(b*x**(1/3)+a*x)**(3/2),x)
Output:
Integral(1/(x**4*(a*x + b*x**(1/3))**(3/2)), x)
\[ \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^(1/3)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((a*x + b*x^(1/3))^(3/2)*x^4), x)
\[ \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^(1/3)+a*x)^(3/2),x, algorithm="giac")
Output:
integrate(1/((a*x + b*x^(1/3))^(3/2)*x^4), x)
Timed out. \[ \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^4\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \] Input:
int(1/(x^4*(a*x + b*x^(1/3))^(3/2)),x)
Output:
int(1/(x^4*(a*x + b*x^(1/3))^(3/2)), x)
\[ \int \frac {1}{x^4 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{\frac {13}{3}} \sqrt {x^{\frac {1}{3}} b +a x}\, b +\sqrt {x^{\frac {1}{3}} b +a x}\, a \,x^{5}}d x \] Input:
int(1/x^4/(b*x^(1/3)+a*x)^(3/2),x)
Output:
int(1/(x**(1/3)*sqrt(x**(1/3)*b + a*x)*b*x**4 + sqrt(x**(1/3)*b + a*x)*a*x **5),x)