Optimal. Leaf size=124 \[ -\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {2 b^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3}}-\frac {b^2 \log (x)}{9 a^{2/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {43, 59, 631,
210, 31} \begin {gather*} -\frac {b^2 \log (x)}{9 a^{2/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}-\frac {2 b^2 \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3}}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {2 b \sqrt [3]{a+b x}}{3 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 43
Rule 59
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {(a+b x)^{4/3}}{x^3} \, dx &=-\frac {(a+b x)^{4/3}}{2 x^2}+\frac {1}{3} (2 b) \int \frac {\sqrt [3]{a+b x}}{x^2} \, dx\\ &=-\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}+\frac {1}{9} \left (2 b^2\right ) \int \frac {1}{x (a+b x)^{2/3}} \, dx\\ &=-\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {b^2 \log (x)}{9 a^{2/3}}-\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{3 a^{2/3}}-\frac {b^2 \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{3 \sqrt [3]{a}}\\ &=-\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {b^2 \log (x)}{9 a^{2/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{3 a^{2/3}}\\ &=-\frac {2 b \sqrt [3]{a+b x}}{3 x}-\frac {(a+b x)^{4/3}}{2 x^2}-\frac {2 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{2/3}}-\frac {b^2 \log (x)}{9 a^{2/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.20, size = 136, normalized size = 1.10 \begin {gather*} \frac {1}{18} \left (-\frac {3 \sqrt [3]{a+b x} (3 a+7 b x)}{x^2}-\frac {4 \sqrt {3} b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {4 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{a^{2/3}}-\frac {2 b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{a^{2/3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 21.91, size = 134, normalized size = 1.08 \begin {gather*} -\frac {a b^{\frac {1}{3}} \left (\frac {a}{b}+x\right )^{\frac {1}{3}}}{2 x^2}-\frac {b^2 \text {Log}\left [1-\frac {b^{\frac {1}{3}} \left (\frac {a}{b}+x\right )^{\frac {1}{3}} \text {exp\_polar}\left [\frac {2 I}{3} \text {Pi}\right ]}{a^{\frac {1}{3}}}\right ]}{9 a^{\frac {2}{3}}}-\frac {b^2 \text {Log}\left [1-\frac {b^{\frac {1}{3}} \left (\frac {a}{b}+x\right )^{\frac {1}{3}} \text {exp\_polar}\left [\frac {4 I}{3} \text {Pi}\right ]}{a^{\frac {1}{3}}}\right ]}{9 a^{\frac {2}{3}}}+\frac {2 b^2 \text {Log}\left [1-\frac {b^{\frac {1}{3}} \left (\frac {a}{b}+x\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}\right ]}{9 a^{\frac {2}{3}}}-\frac {7 b^{\frac {4}{3}} \left (\frac {a}{b}+x\right )^{\frac {1}{3}}}{6 x} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 110, normalized size = 0.89
method | result | size |
derivativedivides | \(3 b^{2} \left (-\frac {\frac {7 \left (b x +a \right )^{\frac {4}{3}}}{18}-\frac {2 a \left (b x +a \right )^{\frac {1}{3}}}{9}}{b^{2} x^{2}}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{27 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{27 a^{\frac {2}{3}}}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{27 a^{\frac {2}{3}}}\right )\) | \(110\) |
default | \(3 b^{2} \left (-\frac {\frac {7 \left (b x +a \right )^{\frac {4}{3}}}{18}-\frac {2 a \left (b x +a \right )^{\frac {1}{3}}}{9}}{b^{2} x^{2}}+\frac {2 \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{27 a^{\frac {2}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{27 a^{\frac {2}{3}}}-\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{27 a^{\frac {2}{3}}}\right )\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.36, size = 136, normalized size = 1.10 \begin {gather*} -\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {2}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {2}{3}}} + \frac {2 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {2}{3}}} - \frac {7 \, {\left (b x + a\right )}^{\frac {4}{3}} b^{2} - 4 \, {\left (b x + a\right )}^{\frac {1}{3}} a b^{2}}{6 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.31, size = 162, normalized size = 1.31 \begin {gather*} -\frac {4 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a b^{2} x^{2} \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\left (\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right )}}{3 \, a^{2}}\right ) + 2 \, {\left (a^{2}\right )}^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (a^{2}\right )}^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}}\right ) - 4 \, {\left (a^{2}\right )}^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 3 \, {\left (7 \, a^{2} b x + 3 \, a^{3}\right )} {\left (b x + a\right )}^{\frac {1}{3}}}{18 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.60, size = 2266, normalized size = 18.27
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.01, size = 198, normalized size = 1.60 \begin {gather*} \frac {-\frac {\frac {1}{9} b^{3} \ln \left (\left (\left (a+b x\right )^{\frac {1}{3}}\right )^{2}+a^{\frac {1}{3}} \left (a+b x\right )^{\frac {1}{3}}+a^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{\left (a^{\frac {1}{3}}\right )^{2}}-\frac {\frac {1}{3}\cdot 2 b^{3} \arctan \left (\frac {2 \left (\left (a+b x\right )^{\frac {1}{3}}+\frac {a^{\frac {1}{3}}}{2}\right )}{\sqrt {3} a^{\frac {1}{3}}}\right )}{\sqrt {3} \left (a^{\frac {1}{3}}\right )^{2}}+\frac {2 b^{3} a^{\frac {1}{3}} \ln \left |\left (a+b x\right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right |}{3\cdot 3 a}+\frac {\frac {1}{6} \left (-7 \left (a+b x\right )^{\frac {1}{3}} \left (a+b x\right ) b^{3}+4 \left (a+b x\right )^{\frac {1}{3}} a b^{3}\right )}{\left (a+b x-a\right )^{2}}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.12, size = 174, normalized size = 1.40 \begin {gather*} \frac {2\,b^2\,\ln \left (2\,b^2\,{\left (a+b\,x\right )}^{1/3}-2\,a^{1/3}\,b^2\right )}{9\,a^{2/3}}-\frac {\frac {7\,b^2\,{\left (a+b\,x\right )}^{4/3}}{6}-\frac {2\,a\,b^2\,{\left (a+b\,x\right )}^{1/3}}{3}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}-\frac {\ln \left (2\,b^2\,{\left (a+b\,x\right )}^{1/3}+a^{1/3}\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )\right )\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}{9\,a^{2/3}}+\frac {b^2\,\ln \left (2\,b^2\,{\left (a+b\,x\right )}^{1/3}-9\,a^{1/3}\,b^2\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}{a^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________