Integrand size = 13, antiderivative size = 109 \[ \int \frac {1}{x^{4/3} (a+b x)} \, dx=-\frac {3}{a \sqrt [3]{x}}+\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{4/3}}+\frac {3 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 a^{4/3}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 a^{4/3}} \]
-3/a/x^(1/3)+3/2*b^(1/3)*ln(a^(1/3)+b^(1/3)*x^(1/3))/a^(4/3)-1/2*b^(1/3)*l n(b*x+a)/a^(4/3)+b^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x^(1/3))/a^(1/3)*3^ (1/2))*3^(1/2)/a^(4/3)
Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^{4/3} (a+b x)} \, dx=\frac {-\frac {6 \sqrt [3]{a}}{\sqrt [3]{x}}+2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt [3]{x}+b^{2/3} x^{2/3}\right )}{2 a^{4/3}} \]
((-6*a^(1/3))/x^(1/3) + 2*Sqrt[3]*b^(1/3)*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/ a^(1/3))/Sqrt[3]] + 2*b^(1/3)*Log[a^(1/3) + b^(1/3)*x^(1/3)] - b^(1/3)*Log [a^(2/3) - a^(1/3)*b^(1/3)*x^(1/3) + b^(2/3)*x^(2/3)])/(2*a^(4/3))
Time = 0.20 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {61, 68, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {1}{x^{4/3} (a+b x)} \, dx\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {b \int \frac {1}{\sqrt [3]{x} (a+b x)}dx}{a}-\frac {3}{a \sqrt [3]{x}}\) |
| \(\Big \downarrow \) 68 |
| \(\displaystyle -\frac {b \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+\sqrt [3]{x}}d\sqrt [3]{x}}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {3}{a \sqrt [3]{x}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {b \left (\frac {3 \int \frac {1}{\frac {a^{2/3}}{b^{2/3}}-\frac {\sqrt [3]{x} \sqrt [3]{a}}{\sqrt [3]{b}}+x^{2/3}}d\sqrt [3]{x}}{2 b}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {3}{a \sqrt [3]{x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {b \left (\frac {3 \int \frac {1}{-x^{2/3}-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {3}{a \sqrt [3]{x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {b \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a} b^{2/3}}-\frac {3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt [3]{x}\right )}{2 \sqrt [3]{a} b^{2/3}}+\frac {\log (a+b x)}{2 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {3}{a \sqrt [3]{x}}\) |
-3/(a*x^(1/3)) - (b*(-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^(1/3))/a^(1/3))/S qrt[3]])/(a^(1/3)*b^(2/3))) - (3*Log[a^(1/3) + b^(1/3)*x^(1/3)])/(2*a^(1/3 )*b^(2/3)) + Log[a + b*x]/(2*a^(1/3)*b^(2/3))))/a
3.7.80.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
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_) + (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]
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(-\frac {3}{a \,x^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 a \left (\frac {a}{b}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}}\) | \(104\) |
| derivativedivides | \(-\frac {3}{a \,x^{\frac {1}{3}}}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(112\) |
| default | \(-\frac {3}{a \,x^{\frac {1}{3}}}-\frac {3 \left (-\frac {\ln \left (x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{\frac {2}{3}}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{\frac {1}{3}}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{\frac {1}{3}}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b}{a}\) | \(112\) |
-3/a/x^(1/3)+1/a/(a/b)^(1/3)*ln(x^(1/3)+(a/b)^(1/3))-1/2/a/(a/b)^(1/3)*ln( x^(2/3)-(a/b)^(1/3)*x^(1/3)+(a/b)^(2/3))-1/a*3^(1/2)/(a/b)^(1/3)*arctan(1/ 3*3^(1/2)*(2/(a/b)^(1/3)*x^(1/3)-1))
Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^{4/3} (a+b x)} \, dx=-\frac {2 \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (-a x^{\frac {1}{3}} \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, x \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (a \left (\frac {b}{a}\right )^{\frac {2}{3}} + b x^{\frac {1}{3}}\right ) + 6 \, x^{\frac {2}{3}}}{2 \, a x} \]
-1/2*(2*sqrt(3)*x*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x^(1/3)*(b/a)^(1/3) - 1/3 *sqrt(3)) + x*(b/a)^(1/3)*log(-a*x^(1/3)*(b/a)^(2/3) + b*x^(2/3) + a*(b/a) ^(1/3)) - 2*x*(b/a)^(1/3)*log(a*(b/a)^(2/3) + b*x^(1/3)) + 6*x^(2/3))/(a*x )
Time = 31.34 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^{4/3} (a+b x)} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {4}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{4 b x^{\frac {4}{3}}} & \text {for}\: a = 0 \\- \frac {3}{a \sqrt [3]{x}} & \text {for}\: b = 0 \\- \frac {\log {\left (\sqrt [3]{x} - \sqrt [3]{- \frac {a}{b}} \right )}}{a \sqrt [3]{- \frac {a}{b}}} + \frac {\log {\left (4 x^{\frac {2}{3}} + 4 \sqrt [3]{x} \sqrt [3]{- \frac {a}{b}} + 4 \left (- \frac {a}{b}\right )^{\frac {2}{3}} \right )}}{2 a \sqrt [3]{- \frac {a}{b}}} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \sqrt [3]{x}}{3 \sqrt [3]{- \frac {a}{b}}} + \frac {\sqrt {3}}{3} \right )}}{a \sqrt [3]{- \frac {a}{b}}} - \frac {3}{a \sqrt [3]{x}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(4/3), Eq(a, 0) & Eq(b, 0)), (-3/(4*b*x**(4/3)), Eq(a, 0 )), (-3/(a*x**(1/3)), Eq(b, 0)), (-log(x**(1/3) - (-a/b)**(1/3))/(a*(-a/b) **(1/3)) + log(4*x**(2/3) + 4*x**(1/3)*(-a/b)**(1/3) + 4*(-a/b)**(2/3))/(2 *a*(-a/b)**(1/3)) - sqrt(3)*atan(2*sqrt(3)*x**(1/3)/(3*(-a/b)**(1/3)) + sq rt(3)/3)/(a*(-a/b)**(1/3)) - 3/(a*x**(1/3)), True))
Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^{4/3} (a+b x)} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{\frac {2}{3}} - x^{\frac {1}{3}} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {3}{a x^{\frac {1}{3}}} \]
-sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/3) - (a/b)^(1/3))/(a/b)^(1/3))/(a*(a/b )^(1/3)) - 1/2*log(x^(2/3) - x^(1/3)*(a/b)^(1/3) + (a/b)^(2/3))/(a*(a/b)^( 1/3)) + log(x^(1/3) + (a/b)^(1/3))/(a*(a/b)^(1/3)) - 3/(a*x^(1/3))
Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^{4/3} (a+b x)} \, dx=\frac {b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{\frac {1}{3}} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{2} b} - \frac {3}{a x^{\frac {1}{3}}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{\frac {2}{3}} + x^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \, a^{2} b} \]
b*(-a/b)^(2/3)*log(abs(x^(1/3) - (-a/b)^(1/3)))/a^2 + sqrt(3)*(-a*b^2)^(2/ 3)*arctan(1/3*sqrt(3)*(2*x^(1/3) + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b) - 3 /(a*x^(1/3)) - 1/2*(-a*b^2)^(2/3)*log(x^(2/3) + x^(1/3)*(-a/b)^(1/3) + (-a /b)^(2/3))/(a^2*b)
Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^{4/3} (a+b x)} \, dx=\frac {b^{1/3}\,\ln \left (9\,a^{4/3}\,b^{8/3}+9\,a\,b^3\,x^{1/3}\right )}{a^{4/3}}-\frac {3}{a\,x^{1/3}}+\frac {b^{1/3}\,\ln \left (9\,a\,b^3\,x^{1/3}+9\,a^{4/3}\,b^{8/3}\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{4/3}}-\frac {b^{1/3}\,\ln \left (9\,a\,b^3\,x^{1/3}+9\,a^{4/3}\,b^{8/3}\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{4/3}} \]
(b^(1/3)*log(9*a^(4/3)*b^(8/3) + 9*a*b^3*x^(1/3)))/a^(4/3) - 3/(a*x^(1/3)) + (b^(1/3)*log(9*a*b^3*x^(1/3) + 9*a^(4/3)*b^(8/3)*((3^(1/2)*1i)/2 - 1/2) ^2)*((3^(1/2)*1i)/2 - 1/2))/a^(4/3) - (b^(1/3)*log(9*a*b^3*x^(1/3) + 9*a^( 4/3)*b^(8/3)*((3^(1/2)*1i)/2 + 1/2)^2)*((3^(1/2)*1i)/2 + 1/2))/a^(4/3)