Optimal. Leaf size=202 \[ -\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3} d}+\frac {2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac {(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]
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Rubi [A] time = 0.17, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {372, 290, 292, 31, 634, 617, 204, 628} \[ -\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}-\frac {2 \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3} d}+\frac {2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac {(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 290
Rule 292
Rule 372
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{\left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 \operatorname {Subst}\left (\int \frac {x}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 a d}\\ &=\frac {(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{9 a^2 d}\\ &=\frac {(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{7/3} \sqrt [3]{b} d}+\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{27 a^{7/3} \sqrt [3]{b} d}\\ &=\frac {(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {\operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{27 a^{7/3} b^{2/3} d}+\frac {\operatorname {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{9 a^2 \sqrt [3]{b} d}\\ &=\frac {(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{9 a^{7/3} b^{2/3} d}\\ &=\frac {(c+d x)^2}{6 a d \left (a+b (c+d x)^3\right )^2}+\frac {2 (c+d x)^2}{9 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3} d}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{7/3} b^{2/3} d}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{7/3} b^{2/3} d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 180, normalized size = 0.89 \[ \frac {\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}+\frac {9 a^{4/3} (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac {4 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}+\frac {12 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}}{54 a^{7/3} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 1708, normalized size = 8.46 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 268, normalized size = 1.33 \[ -\frac {2 \, \sqrt {3} \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{27 \, a^{2}} + \frac {4 \, b d^{5} x^{5} + 20 \, b c d^{4} x^{4} + 40 \, b c^{2} d^{3} x^{3} + 40 \, b c^{3} d^{2} x^{2} + 20 \, b c^{4} d x + 4 \, b c^{5} + 7 \, a d^{2} x^{2} + 14 \, a c d x + 7 \, a c^{2}}{18 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 214, normalized size = 1.06 \[ \frac {2 \left (\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right ) d +c \right ) \ln \left (-\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+x \right )}{27 a^{2} b d \left (d^{2} \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )^{2}+2 c d \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+c^{2}\right )}+\frac {\frac {2 b \,d^{4} x^{5}}{9 a^{2}}+\frac {10 b c \,d^{3} x^{4}}{9 a^{2}}+\frac {20 b \,c^{2} d^{2} x^{3}}{9 a^{2}}+\frac {\left (40 b \,c^{3}+7 a \right ) d \,x^{2}}{18 a^{2}}+\frac {\left (10 b \,c^{3}+7 a \right ) c x}{9 a^{2}}+\frac {\left (4 b \,c^{3}+7 a \right ) c^{2}}{18 a^{2} d}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, b d^{5} x^{5} + 20 \, b c d^{4} x^{4} + 40 \, b c^{2} d^{3} x^{3} + 4 \, b c^{5} + {\left (40 \, b c^{3} + 7 \, a\right )} d^{2} x^{2} + 7 \, a c^{2} + 2 \, {\left (10 \, b c^{4} + 7 \, a c\right )} d x}{18 \, {\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \, {\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \, {\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x + {\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )}} + \frac {2 \, {\left (-\frac {1}{3} \, \sqrt {3} \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) - \frac {1}{6} \, \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) + \frac {1}{3} \, \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )\right )}}{9 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 403, normalized size = 2.00 \[ \frac {\frac {4\,b\,c^5+7\,a\,c^2}{18\,a^2\,d}+\frac {2\,b\,d^4\,x^5}{9\,a^2}+\frac {c\,x\,\left (10\,b\,c^3+7\,a\right )}{9\,a^2}+\frac {d\,x^2\,\left (40\,b\,c^3+7\,a\right )}{18\,a^2}+\frac {10\,b\,c\,d^3\,x^4}{9\,a^2}+\frac {20\,b\,c^2\,d^2\,x^3}{9\,a^2}}{x^3\,\left (20\,b^2\,c^3\,d^3+2\,a\,b\,d^3\right )+x^2\,\left (15\,b^2\,c^4\,d^2+6\,a\,b\,c\,d^2\right )+a^2+x\,\left (6\,d\,b^2\,c^5+6\,a\,d\,b\,c^2\right )+b^2\,c^6+b^2\,d^6\,x^6+2\,a\,b\,c^3+6\,b^2\,c\,d^5\,x^5+15\,b^2\,c^2\,d^4\,x^4}+\frac {2\,\ln \left (\frac {4\,b\,c\,d^4}{81\,a^4}-\frac {4\,b^{2/3}\,d^4}{81\,{\left (-a\right )}^{11/3}}+\frac {4\,b\,d^5\,x}{81\,a^4}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}\,d}+\frac {\ln \left (\frac {4\,b\,c\,d^4}{81\,a^4}-\frac {b^{2/3}\,d^4\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{81\,{\left (-a\right )}^{11/3}}+\frac {4\,b\,d^5\,x}{81\,a^4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}\,d}-\frac {\ln \left (\frac {4\,b\,c\,d^4}{81\,a^4}-\frac {b^{2/3}\,d^4\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{81\,{\left (-a\right )}^{11/3}}+\frac {4\,b\,d^5\,x}{81\,a^4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.28, size = 296, normalized size = 1.47 \[ \frac {7 a c^{2} + 4 b c^{5} + 40 b c^{2} d^{3} x^{3} + 20 b c d^{4} x^{4} + 4 b d^{5} x^{5} + x^{2} \left (7 a d^{2} + 40 b c^{3} d^{2}\right ) + x \left (14 a c d + 20 b c^{4} d\right )}{18 a^{4} d + 36 a^{3} b c^{3} d + 18 a^{2} b^{2} c^{6} d + 270 a^{2} b^{2} c^{2} d^{5} x^{4} + 108 a^{2} b^{2} c d^{6} x^{5} + 18 a^{2} b^{2} d^{7} x^{6} + x^{3} \left (36 a^{3} b d^{4} + 360 a^{2} b^{2} c^{3} d^{4}\right ) + x^{2} \left (108 a^{3} b c d^{3} + 270 a^{2} b^{2} c^{4} d^{3}\right ) + x \left (108 a^{3} b c^{2} d^{2} + 108 a^{2} b^{2} c^{5} d^{2}\right )} + \frac {\operatorname {RootSum} {\left (19683 t^{3} a^{7} b^{2} + 8, \left (t \mapsto t \log {\left (x + \frac {729 t^{2} a^{5} b + 4 c}{4 d} \right )} \right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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