Optimal. Leaf size=189 \[ -\frac {2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 a^{7/3} d}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{7/3} d}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} d}-\frac {4}{3 a^2 d (c+d x)}+\frac {1}{3 a d (c+d x) \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.16, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {372, 290, 325, 292, 31, 634, 617, 204, 628} \[ -\frac {2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 a^{7/3} d}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{7/3} d}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3} d}-\frac {4}{3 a^2 d (c+d x)}+\frac {1}{3 a d (c+d x) \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 290
Rule 292
Rule 325
Rule 372
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {1}{3 a d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {4 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{3 a d}\\ &=-\frac {4}{3 a^2 d (c+d x)}+\frac {1}{3 a d (c+d x) \left (a+b (c+d x)^3\right )}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {x}{a+b x^3} \, dx,x,c+d x\right )}{3 a^2 d}\\ &=-\frac {4}{3 a^2 d (c+d x)}+\frac {1}{3 a d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {\left (4 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{9 a^{7/3} d}-\frac {\left (4 b^{2/3}\right ) \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 )}{9 a^{7/3} d}\\ &=-\frac {4}{3 a^2 d (c+d x)}+\frac {1}{3 a d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{7/3} d}-\frac {\left (2 \sqrt [3]{b}\right ) \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 )}{9 a^{7/3} d}-\frac {\left (2 b^{2/3}\right ) \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 )}{3 a^2 d}\\ &=-\frac {4}{3 a^2 d (c+d x)}+\frac {1}{3 a d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{7/3} d}-\frac {2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 a^{7/3} d}-\frac {\left (4 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{7/3} d}\\ &=-\frac {4}{3 a^2 d (c+d x)}+\frac {1}{3 a d (c+d x) \left (a+b (c+d x)^3\right )}+\frac {4 \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{7/3} d}+\frac {4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{7/3} d}-\frac {2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 a^{7/3} d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 168, normalized size = 0.89 \[ \frac {-2 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac {3 \sqrt [3]{a} b (c+d x)^2}{a+b (c+d x)^3}+4 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-4 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )-\frac {9 \sqrt [3]{a}}{c+d x}}{9 a^{7/3} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 384, normalized size = 2.03 \[ -\frac {12 \, b d^{3} x^{3} + 36 \, b c d^{2} x^{2} + 36 \, b c^{2} d x + 12 \, b c^{3} + 4 \, \sqrt {3} {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + b c^{4} + {\left (4 \, b c^{3} + a\right )} d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + b c^{4} + {\left (4 \, b c^{3} + a\right )} d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - {\left (a d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 4 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + b c^{4} + {\left (4 \, b c^{3} + a\right )} d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d x + b c + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 9 \, a}{9 \, {\left (a^{2} b d^{5} x^{4} + 4 \, a^{2} b c d^{4} x^{3} + 6 \, a^{2} b c^{2} d^{3} x^{2} + {\left (4 \, a^{2} b c^{3} + a^{3}\right )} d^{2} x + {\left (a^{2} b c^{4} + a^{3} c\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 204, normalized size = 1.08 \[ \frac {4 \, \left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | -\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} - \frac {1}{{\left (d x + c\right )} d} \right |}\right )}{9 \, a^{2}} - \frac {4 \, \sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} - \frac {2}{{\left (d x + c\right )} d}\right )}}{3 \, \left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} d} - \frac {2 \, \left (a^{2} b\right )^{\frac {1}{3}} \log \left (\left (\frac {b}{a d^{3}}\right )^{\frac {2}{3}} - \frac {\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}{{\left (d x + c\right )} d} + \frac {1}{{\left (d x + c\right )}^{2} d^{2}}\right )}{9 \, a^{3} d} - \frac {1}{{\left (d x + c\right )} a^{2} d} - \frac {b}{3 \, {\left (d x + c\right )} a^{2} {\left (b + \frac {a}{{\left (d x + c\right )}^{3}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 227, normalized size = 1.20 \[ -\frac {b d \,x^{2}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a^{2}}-\frac {2 b c x}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a^{2}}-\frac {b \,c^{2}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a^{2} d}-\frac {4 \left (\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right ) d +c \right ) \ln \left (-\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+x \right )}{9 a^{2} d \left (d^{2} \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )^{2}+2 c d \RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b \,d^{2} c \,\textit {\_Z}^{2}+3 b d \,c^{2} \textit {\_Z} +b \,c^{3}+a \right )+c^{2}\right )}-\frac {1}{\left (d x +c \right ) a^{2} d} \]
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 \[ -\frac {4 \, b d^{3} x^{3} + 12 \, b c d^{2} x^{2} + 12 \, b c^{2} d x + 4 \, b c^{3} + 3 \, a}{3 \, {\left (a^{2} b d^{5} x^{4} + 4 \, a^{2} b c d^{4} x^{3} + 6 \, a^{2} b c^{2} d^{3} x^{2} + {\left (4 \, a^{2} b c^{3} + a^{3}\right )} d^{2} x + {\left (a^{2} b c^{4} + a^{3} c\right )} d\right )}} - \frac {-\frac {2}{3} \, {\left (2 \, \sqrt {3} \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )\right )} b}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.81, size = 231, normalized size = 1.22 \[ \frac {4\,b^{1/3}\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{9\,a^{7/3}\,d}-\frac {\frac {4\,b\,c^3+3\,a}{3\,a^2\,d}+\frac {4\,b\,d^2\,x^3}{3\,a^2}+\frac {4\,b\,c^2\,x}{a^2}+\frac {4\,b\,c\,d\,x^2}{a^2}}{a\,c+x\,\left (4\,b\,d\,c^3+a\,d\right )+b\,c^4+b\,d^4\,x^4+6\,b\,c^2\,d^2\,x^2+4\,b\,c\,d^3\,x^3}-\frac {4\,b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{7/3}\,d}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {2}{9}+\frac {\sqrt {3}\,2{}\mathrm {i}}{9}\right )}{a^{7/3}\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.95, size = 170, normalized size = 0.90 \[ \frac {- 3 a - 4 b c^{3} - 12 b c^{2} d x - 12 b c d^{2} x^{2} - 4 b d^{3} x^{3}}{3 a^{3} c d + 3 a^{2} b c^{4} d + 18 a^{2} b c^{2} d^{3} x^{2} + 12 a^{2} b c d^{4} x^{3} + 3 a^{2} b d^{5} x^{4} + x \left (3 a^{3} d^{2} + 12 a^{2} b c^{3} d^{2}\right )} + \frac {\operatorname {RootSum} {\left (729 t^{3} a^{7} - 64 b, \left (t \mapsto t \log {\left (x + \frac {81 t^{2} a^{5} + 16 b c}{16 b d} \right )} \right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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