Optimal. Leaf size=113 \[ \frac {3 b^7}{a^8 \left (b+a \sqrt [3]{x}\right )}-\frac {18 b^5 \sqrt [3]{x}}{a^7}+\frac {15 b^4 x^{2/3}}{2 a^6}-\frac {4 b^3 x}{a^5}+\frac {9 b^2 x^{4/3}}{4 a^4}-\frac {6 b x^{5/3}}{5 a^3}+\frac {x^2}{2 a^2}+\frac {21 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^8} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 45}
\begin {gather*} \frac {3 b^7}{a^8 \left (a \sqrt [3]{x}+b\right )}+\frac {21 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^8}-\frac {18 b^5 \sqrt [3]{x}}{a^7}+\frac {15 b^4 x^{2/3}}{2 a^6}-\frac {4 b^3 x}{a^5}+\frac {9 b^2 x^{4/3}}{4 a^4}-\frac {6 b x^{5/3}}{5 a^3}+\frac {x^2}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {b}{\sqrt [3]{x}}\right )^2} \, dx &=\int \frac {x^{5/3}}{\left (b+a \sqrt [3]{x}\right )^2} \, dx\\ &=3 \text {Subst}\left (\int \frac {x^7}{(b+a x)^2} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (-\frac {6 b^5}{a^7}+\frac {5 b^4 x}{a^6}-\frac {4 b^3 x^2}{a^5}+\frac {3 b^2 x^3}{a^4}-\frac {2 b x^4}{a^3}+\frac {x^5}{a^2}-\frac {b^7}{a^7 (b+a x)^2}+\frac {7 b^6}{a^7 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 b^7}{a^8 \left (b+a \sqrt [3]{x}\right )}-\frac {18 b^5 \sqrt [3]{x}}{a^7}+\frac {15 b^4 x^{2/3}}{2 a^6}-\frac {4 b^3 x}{a^5}+\frac {9 b^2 x^{4/3}}{4 a^4}-\frac {6 b x^{5/3}}{5 a^3}+\frac {x^2}{2 a^2}+\frac {21 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^8}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 121, normalized size = 1.07 \begin {gather*} \frac {60 b^7-360 a b^6 \sqrt [3]{x}-210 a^2 b^5 x^{2/3}+70 a^3 b^4 x-35 a^4 b^3 x^{4/3}+21 a^5 b^2 x^{5/3}-14 a^6 b x^2+10 a^7 x^{7/3}}{20 a^8 \left (b+a \sqrt [3]{x}\right )}+\frac {21 b^6 \log \left (b+a \sqrt [3]{x}\right )}{a^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.20, size = 95, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {a^{5} x^{2}}{2}-\frac {6 a^{4} b \,x^{\frac {5}{3}}}{5}+\frac {9 a^{3} b^{2} x^{\frac {4}{3}}}{4}-4 a^{2} b^{3} x +\frac {15 a \,b^{4} x^{\frac {2}{3}}}{2}-18 b^{5} x^{\frac {1}{3}}}{a^{7}}+\frac {3 b^{7}}{a^{8} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {21 b^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{8}}\) | \(95\) |
default | \(\frac {\frac {a^{5} x^{2}}{2}-\frac {6 a^{4} b \,x^{\frac {5}{3}}}{5}+\frac {9 a^{3} b^{2} x^{\frac {4}{3}}}{4}-4 a^{2} b^{3} x +\frac {15 a \,b^{4} x^{\frac {2}{3}}}{2}-18 b^{5} x^{\frac {1}{3}}}{a^{7}}+\frac {3 b^{7}}{a^{8} \left (b +a \,x^{\frac {1}{3}}\right )}+\frac {21 b^{6} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{a^{8}}\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 112, normalized size = 0.99 \begin {gather*} \frac {10 \, a^{6} - \frac {14 \, a^{5} b}{x^{\frac {1}{3}}} + \frac {21 \, a^{4} b^{2}}{x^{\frac {2}{3}}} - \frac {35 \, a^{3} b^{3}}{x} + \frac {70 \, a^{2} b^{4}}{x^{\frac {4}{3}}} - \frac {210 \, a b^{5}}{x^{\frac {5}{3}}} - \frac {420 \, b^{6}}{x^{2}}}{20 \, {\left (\frac {a^{8}}{x^{2}} + \frac {a^{7} b}{x^{\frac {7}{3}}}\right )}} + \frac {21 \, b^{6} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{a^{8}} + \frac {7 \, b^{6} \log \left (x\right )}{a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 137, normalized size = 1.21 \begin {gather*} \frac {10 \, a^{9} x^{3} - 70 \, a^{6} b^{3} x^{2} - 80 \, a^{3} b^{6} x + 60 \, b^{9} + 420 \, {\left (a^{3} b^{6} x + b^{9}\right )} \log \left (a x^{\frac {1}{3}} + b\right ) - 6 \, {\left (4 \, a^{8} b x^{2} - 21 \, a^{5} b^{4} x - 35 \, a^{2} b^{7}\right )} x^{\frac {2}{3}} + 15 \, {\left (3 \, a^{7} b^{2} x^{2} - 21 \, a^{4} b^{5} x - 28 \, a b^{8}\right )} x^{\frac {1}{3}}}{20 \, {\left (a^{11} x + a^{8} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 277 vs.
\(2 (110) = 220\).
time = 0.46, size = 277, normalized size = 2.45 \begin {gather*} \begin {cases} \frac {10 a^{7} x^{\frac {7}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac {14 a^{6} b x^{2}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {21 a^{5} b^{2} x^{\frac {5}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac {35 a^{4} b^{3} x^{\frac {4}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {70 a^{3} b^{4} x}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} - \frac {210 a^{2} b^{5} x^{\frac {2}{3}}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {420 a b^{6} \sqrt [3]{x} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {420 b^{7} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} + \frac {420 b^{7}}{20 a^{9} \sqrt [3]{x} + 20 a^{8} b} & \text {for}\: a \neq 0 \\\frac {3 x^{\frac {8}{3}}}{8 b^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.58, size = 100, normalized size = 0.88 \begin {gather*} \frac {21 \, b^{6} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{a^{8}} + \frac {3 \, b^{7}}{{\left (a x^{\frac {1}{3}} + b\right )} a^{8}} + \frac {10 \, a^{10} x^{2} - 24 \, a^{9} b x^{\frac {5}{3}} + 45 \, a^{8} b^{2} x^{\frac {4}{3}} - 80 \, a^{7} b^{3} x + 150 \, a^{6} b^{4} x^{\frac {2}{3}} - 360 \, a^{5} b^{5} x^{\frac {1}{3}}}{20 \, a^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.13, size = 99, normalized size = 0.88 \begin {gather*} \frac {x^2}{2\,a^2}+\frac {3\,b^7}{a\,\left (a^7\,b+a^8\,x^{1/3}\right )}-\frac {4\,b^3\,x}{a^5}-\frac {6\,b\,x^{5/3}}{5\,a^3}+\frac {21\,b^6\,\ln \left (b+a\,x^{1/3}\right )}{a^8}+\frac {9\,b^2\,x^{4/3}}{4\,a^4}+\frac {15\,b^4\,x^{2/3}}{2\,a^6}-\frac {18\,b^5\,x^{1/3}}{a^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________