Integrand size = 11, antiderivative size = 184 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx=\frac {2 x}{9 a \left (a+b x^{3/2}\right )^3}+\frac {7 x}{27 a^2 \left (a+b x^{3/2}\right )^2}+\frac {28 x}{81 a^3 \left (a+b x^{3/2}\right )}-\frac {28 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{10/3} b^{2/3}}-\frac {28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{243 a^{10/3} b^{2/3}}+\frac {14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{243 a^{10/3} b^{2/3}} \] Output:
2/9*x/a/(a+b*x^(3/2))^3+7/27*x/a^2/(a+b*x^(3/2))^2+28/81*x/a^3/(a+b*x^(3/2 ))-28/243*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/2))*3^(1/2)/a^(1/3))*3^(1/2)/ a^(10/3)/b^(2/3)-28/243*ln(a^(1/3)+b^(1/3)*x^(1/2))/a^(10/3)/b^(2/3)+14/24 3*ln(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/2)+b^(2/3)*x)/a^(10/3)/b^(2/3)
Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx=\frac {\frac {3 \sqrt [3]{a} x \left (67 a^2+77 a b x^{3/2}+28 b^2 x^3\right )}{\left (a+b x^{3/2}\right )^3}-\frac {28 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-\frac {28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{b^{2/3}}+\frac {14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{b^{2/3}}}{243 a^{10/3}} \] Input:
Integrate[(a + b*x^(3/2))^(-4),x]
Output:
((3*a^(1/3)*x*(67*a^2 + 77*a*b*x^(3/2) + 28*b^2*x^3))/(a + b*x^(3/2))^3 - (28*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*Sqrt[x])/a^(1/3))/Sqrt[3]])/b^(2/3) - ( 28*Log[a^(1/3) + b^(1/3)*Sqrt[x]])/b^(2/3) + (14*Log[a^(2/3) - a^(1/3)*b^( 1/3)*Sqrt[x] + b^(2/3)*x])/b^(2/3))/(243*a^(10/3))
Time = 0.57 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.091, Rules used = {774, 819, 819, 819, 821, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx\) |
| \(\Big \downarrow \) 774 |
| \(\displaystyle 2 \int \frac {\sqrt {x}}{\left (b x^{3/2}+a\right )^4}d\sqrt {x}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle 2 \left (\frac {7 \int \frac {\sqrt {x}}{\left (b x^{3/2}+a\right )^3}d\sqrt {x}}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \int \frac {\sqrt {x}}{\left (b x^{3/2}+a\right )^2}d\sqrt {x}}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\int \frac {\sqrt {x}}{b x^{3/2}+a}d\sqrt {x}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}\right )}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}\right )}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {3 \int \frac {1}{-x-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 2 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^{3/2}\right )^3}\right )\) |
Input:
Int[(a + b*x^(3/2))^(-4),x]
Output:
2*(x/(9*a*(a + b*x^(3/2))^3) + (7*(x/(6*a*(a + b*x^(3/2))^2) + (2*(x/(3*a* (a + b*x^(3/2))) + (-1/3*Log[a^(1/3) + b^(1/3)*Sqrt[x]]/(a^(1/3)*b^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*Sqrt[x])/a^(1/3))/Sqrt[3]])/b^(1/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sqrt[x] + b^(2/3)*x]/(2*b^(1/3)))/(3*a^(1/ 3)*b^(1/3)))/(3*a)))/(3*a)))/(9*a))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
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_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(\frac {2 x}{9 a \left (a +b \,x^{\frac {3}{2}}\right )^{3}}+\frac {\frac {7 x}{27 a \left (a +b \,x^{\frac {3}{2}}\right )^{2}}+\frac {14 \left (\frac {2 x}{9 a \left (a +b \,x^{\frac {3}{2}}\right )}+\frac {2 \left (-\frac {\ln \left (\sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a}\right )}{9 a}}{a}\) | \(158\) |
| default | \(\text {Expression too large to display}\) | \(1079\) |
Input:
int(1/(a+b*x^(3/2))^4,x,method=_RETURNVERBOSE)
Output:
2/9*x/a/(a+b*x^(3/2))^3+14/9/a*(1/6*x/a/(a+b*x^(3/2))^2+2/3/a*(1/3*x/a/(a+ b*x^(3/2))+1/3/a*(-1/3/b/(a/b)^(1/3)*ln(x^(1/2)+(a/b)^(1/3))+1/6/b/(a/b)^( 1/3)*ln(x-(a/b)^(1/3)*x^(1/2)+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arcta n(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/2)-1)))))
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (129) = 258\).
Time = 0.11 (sec) , antiderivative size = 787, normalized size of antiderivative = 4.28 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b*x^(3/2))^4,x, algorithm="fricas")
Output:
[-1/243*(21*a^2*b^6*x^7 - 6*a^4*b^4*x^4 + 201*a^6*b^2*x - 42*sqrt(1/3)*(a* b^7*x^9 - 3*a^3*b^5*x^6 + 3*a^5*b^3*x^3 - a^7*b)*sqrt(-(a*b^2)^(1/3)/a)*lo g((2*b^3*x^3 - 3*(a*b^2)^(2/3)*b*x^2 + a^2*b + 3*sqrt(1/3)*(a*b^2*x^2 - 2* (a*b^2)^(2/3)*a*x + (a*b^2)^(1/3)*a^2 + (2*(a*b^2)^(2/3)*b*x^2 - (a*b^2)^( 1/3)*a*b*x - a^2*b)*sqrt(x))*sqrt(-(a*b^2)^(1/3)/a) - 3*(a*b^2*x - (a*b^2) ^(2/3)*a)*sqrt(x))/(b^2*x^3 - a^2)) - 14*(b^6*x^9 - 3*a^2*b^4*x^6 + 3*a^4* b^2*x^3 - a^6)*(a*b^2)^(2/3)*log(b^2*x - (a*b^2)^(1/3)*b*sqrt(x) + (a*b^2) ^(2/3)) + 28*(b^6*x^9 - 3*a^2*b^4*x^6 + 3*a^4*b^2*x^3 - a^6)*(a*b^2)^(2/3) *log(b*sqrt(x) + (a*b^2)^(1/3)) - 12*(7*a*b^7*x^8 - 20*a^3*b^5*x^5 + 31*a^ 5*b^3*x^2)*sqrt(x))/(a^4*b^8*x^9 - 3*a^6*b^6*x^6 + 3*a^8*b^4*x^3 - a^10*b^ 2), -1/243*(21*a^2*b^6*x^7 - 6*a^4*b^4*x^4 + 201*a^6*b^2*x - 84*sqrt(1/3)* (a*b^7*x^9 - 3*a^3*b^5*x^6 + 3*a^5*b^3*x^3 - a^7*b)*sqrt((a*b^2)^(1/3)/a)* arctan(sqrt(1/3)*(2*b*sqrt(x) - (a*b^2)^(1/3))*sqrt((a*b^2)^(1/3)/a)/b) - 14*(b^6*x^9 - 3*a^2*b^4*x^6 + 3*a^4*b^2*x^3 - a^6)*(a*b^2)^(2/3)*log(b^2*x - (a*b^2)^(1/3)*b*sqrt(x) + (a*b^2)^(2/3)) + 28*(b^6*x^9 - 3*a^2*b^4*x^6 + 3*a^4*b^2*x^3 - a^6)*(a*b^2)^(2/3)*log(b*sqrt(x) + (a*b^2)^(1/3)) - 12*( 7*a*b^7*x^8 - 20*a^3*b^5*x^5 + 31*a^5*b^3*x^2)*sqrt(x))/(a^4*b^8*x^9 - 3*a ^6*b^6*x^6 + 3*a^8*b^4*x^3 - a^10*b^2)]
Timed out. \[ \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*x**(3/2))**4,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx=\frac {28 \, b^{2} x^{4} + 77 \, a b x^{\frac {5}{2}} + 67 \, a^{2} x}{81 \, {\left (a^{3} b^{3} x^{\frac {9}{2}} + 3 \, a^{4} b^{2} x^{3} + 3 \, a^{5} b x^{\frac {3}{2}} + a^{6}\right )}} + \frac {28 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {14 \, \log \left (x - \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {28 \, \log \left (\sqrt {x} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(1/(a+b*x^(3/2))^4,x, algorithm="maxima")
Output:
1/81*(28*b^2*x^4 + 77*a*b*x^(5/2) + 67*a^2*x)/(a^3*b^3*x^(9/2) + 3*a^4*b^2 *x^3 + 3*a^5*b*x^(3/2) + a^6) + 28/243*sqrt(3)*arctan(1/3*sqrt(3)*(2*sqrt( x) - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b*(a/b)^(1/3)) + 14/243*log(x - sqrt(x )*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b*(a/b)^(1/3)) - 28/243*log(sqrt(x) + (a /b)^(1/3))/(a^3*b*(a/b)^(1/3))
Time = 0.14 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx=-\frac {28 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{4}} - \frac {28 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{4} b^{2}} + \frac {14 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x + \sqrt {x} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{4} b^{2}} + \frac {28 \, b^{2} x^{4} + 77 \, a b x^{\frac {5}{2}} + 67 \, a^{2} x}{81 \, {\left (b x^{\frac {3}{2}} + a\right )}^{3} a^{3}} \] Input:
integrate(1/(a+b*x^(3/2))^4,x, algorithm="giac")
Output:
-28/243*(-a/b)^(2/3)*log(abs(sqrt(x) - (-a/b)^(1/3)))/a^4 - 28/243*sqrt(3) *(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + (-a/b)^(1/3))/(-a/b)^(1/3) )/(a^4*b^2) + 14/243*(-a*b^2)^(2/3)*log(x + sqrt(x)*(-a/b)^(1/3) + (-a/b)^ (2/3))/(a^4*b^2) + 1/81*(28*b^2*x^4 + 77*a*b*x^(5/2) + 67*a^2*x)/((b*x^(3/ 2) + a)^3*a^3)
Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx=\frac {\frac {67\,x}{81\,a}+\frac {77\,b\,x^{5/2}}{81\,a^2}+\frac {28\,b^2\,x^4}{81\,a^3}}{a^3+b^3\,x^{9/2}+3\,a\,b^2\,x^3+3\,a^2\,b\,x^{3/2}}+\frac {28\,{\left (-1\right )}^{1/3}\,\ln \left (\frac {784\,{\left (-1\right )}^{2/3}\,b^{2/3}}{6561\,a^{17/3}}+\frac {784\,b\,\sqrt {x}}{6561\,a^6}\right )}{243\,a^{10/3}\,b^{2/3}}+\frac {28\,{\left (-1\right )}^{1/3}\,\ln \left (\frac {784\,b\,\sqrt {x}}{6561\,a^6}+\frac {784\,{\left (-1\right )}^{2/3}\,b^{2/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{6561\,a^{17/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{243\,a^{10/3}\,b^{2/3}}-\frac {28\,{\left (-1\right )}^{1/3}\,\ln \left (\frac {784\,b\,\sqrt {x}}{6561\,a^6}+\frac {784\,{\left (-1\right )}^{2/3}\,b^{2/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{6561\,a^{17/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{243\,a^{10/3}\,b^{2/3}} \] Input:
int(1/(a + b*x^(3/2))^4,x)
Output:
((67*x)/(81*a) + (77*b*x^(5/2))/(81*a^2) + (28*b^2*x^4)/(81*a^3))/(a^3 + b ^3*x^(9/2) + 3*a*b^2*x^3 + 3*a^2*b*x^(3/2)) + (28*(-1)^(1/3)*log((784*(-1) ^(2/3)*b^(2/3))/(6561*a^(17/3)) + (784*b*x^(1/2))/(6561*a^6)))/(243*a^(10/ 3)*b^(2/3)) + (28*(-1)^(1/3)*log((784*b*x^(1/2))/(6561*a^6) + (784*(-1)^(2 /3)*b^(2/3)*((3^(1/2)*1i)/2 - 1/2)^2)/(6561*a^(17/3)))*((3^(1/2)*1i)/2 - 1 /2))/(243*a^(10/3)*b^(2/3)) - (28*(-1)^(1/3)*log((784*b*x^(1/2))/(6561*a^6 ) + (784*(-1)^(2/3)*b^(2/3)*((3^(1/2)*1i)/2 + 1/2)^2)/(6561*a^(17/3)))*((3 ^(1/2)*1i)/2 + 1/2))/(243*a^(10/3)*b^(2/3))
Time = 0.20 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.10 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^4} \, dx=\frac {-84 \sqrt {x}\, \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} b x -28 \sqrt {x}\, \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{3} x^{4}-28 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{3}-84 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a \,b^{2} x^{3}+231 \sqrt {x}\, b^{\frac {5}{3}} a^{\frac {4}{3}} x^{2}+201 b^{\frac {2}{3}} a^{\frac {7}{3}} x +84 b^{\frac {8}{3}} a^{\frac {1}{3}} x^{4}+42 \sqrt {x}\, \mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) a^{2} b x +14 \sqrt {x}\, \mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) b^{3} x^{4}-84 \sqrt {x}\, \mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) a^{2} b x -28 \sqrt {x}\, \mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) b^{3} x^{4}+14 \,\mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) a^{3}+42 \,\mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) a \,b^{2} x^{3}-28 \,\mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) a^{3}-84 \,\mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) a \,b^{2} x^{3}}{243 b^{\frac {2}{3}} a^{\frac {10}{3}} \left (3 \sqrt {x}\, a^{2} b x +\sqrt {x}\, b^{3} x^{4}+a^{3}+3 a \,b^{2} x^{3}\right )} \] Input:
int(1/(a+b*x^(3/2))^4,x)
Output:
( - 84*sqrt(x)*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**(1/3))/(a**(1/3)*sqrt (3)))*a**2*b*x - 28*sqrt(x)*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**(1/3))/( a**(1/3)*sqrt(3)))*b**3*x**4 - 28*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**(1 /3))/(a**(1/3)*sqrt(3)))*a**3 - 84*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**( 1/3))/(a**(1/3)*sqrt(3)))*a*b**2*x**3 + 231*sqrt(x)*b**(2/3)*a**(1/3)*a*b* x**2 + 201*b**(2/3)*a**(1/3)*a**2*x + 84*b**(2/3)*a**(1/3)*b**2*x**4 + 42* sqrt(x)*log(a**(2/3) - sqrt(x)*b**(1/3)*a**(1/3) + b**(2/3)*x)*a**2*b*x + 14*sqrt(x)*log(a**(2/3) - sqrt(x)*b**(1/3)*a**(1/3) + b**(2/3)*x)*b**3*x** 4 - 84*sqrt(x)*log(a**(1/3) + sqrt(x)*b**(1/3))*a**2*b*x - 28*sqrt(x)*log( a**(1/3) + sqrt(x)*b**(1/3))*b**3*x**4 + 14*log(a**(2/3) - sqrt(x)*b**(1/3 )*a**(1/3) + b**(2/3)*x)*a**3 + 42*log(a**(2/3) - sqrt(x)*b**(1/3)*a**(1/3 ) + b**(2/3)*x)*a*b**2*x**3 - 28*log(a**(1/3) + sqrt(x)*b**(1/3))*a**3 - 8 4*log(a**(1/3) + sqrt(x)*b**(1/3))*a*b**2*x**3)/(243*b**(2/3)*a**(1/3)*a** 3*(3*sqrt(x)*a**2*b*x + sqrt(x)*b**3*x**4 + a**3 + 3*a*b**2*x**3))