Optimal. Leaf size=219 \[ -\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {20 b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} d}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d} \]
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Rubi [A]
time = 0.13, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {379, 296, 331,
206, 31, 648, 631, 210, 642} \begin {gather*} \frac {20 b^{2/3} \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} d}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}-\frac {10}{9 a^3 d (c+d x)^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 296
Rule 331
Rule 379
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^3 \left (a+b (c+d x)^3\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4 \text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{3 a d}\\ &=\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {20 \text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{9 a^2 d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {(20 b) \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,c+d x\right )}{9 a^3 d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {(20 b) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{27 a^{11/3} d}-\frac {(20 b) \text {Subst}\left (\int \frac {2 \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^{11/3} d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {\left (10 b^{2/3}\right ) \text {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^{11/3} d}-\frac {(10 b) \text {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^{10/3} d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}-\frac {\left (20 b^{2/3}\right ) \text {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^{11/3} d}\\ &=-\frac {10}{9 a^3 d (c+d x)^2}+\frac {1}{6 a d (c+d x)^2 \left (a+b (c+d x)^3\right )^2}+\frac {4}{9 a^2 d (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {20 b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{11/3} d}-\frac {20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d}+\frac {10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 192, normalized size = 0.88 \begin {gather*} \frac {-\frac {27 a^{2/3}}{(c+d x)^2}-\frac {9 a^{5/3} b (c+d x)}{\left (a+b (c+d x)^3\right )^2}-\frac {33 a^{2/3} b (c+d x)}{a+b (c+d x)^3}-40 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-40 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )+20 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{54 a^{11/3} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.26, size = 188, normalized size = 0.86
method | result | size |
default | \(-\frac {1}{2 a^{3} d \left (d x +c \right )^{2}}-\frac {b \left (\frac {\frac {11 b \,d^{3} x^{4}}{18}+\frac {22 b c \,d^{2} x^{3}}{9}+\frac {11 b \,c^{2} d \,x^{2}}{3}+\left (\frac {22 b \,c^{3}}{9}+\frac {7 a}{9}\right ) x +\frac {c \left (11 b \,c^{3}+14 a \right )}{18 d}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2}}+\frac {20 \left (\munderset {\textit {\_R} =\RootOf \left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +b \,c^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{27 b d}\right )}{a^{3}}\) | \(188\) |
risch | \(\frac {-\frac {10 b^{2} d^{5} x^{6}}{9 a^{3}}-\frac {20 b^{2} c \,d^{4} x^{5}}{3 a^{3}}-\frac {50 b^{2} c^{2} d^{3} x^{4}}{3 a^{3}}-\frac {8 b \,d^{2} \left (25 b \,c^{3}+2 a \right ) x^{3}}{9 a^{3}}-\frac {2 b c d \left (25 b \,c^{3}+8 a \right ) x^{2}}{3 a^{3}}-\frac {4 b \,c^{2} \left (5 b \,c^{3}+4 a \right ) x}{3 a^{3}}-\frac {20 b^{2} c^{6}+32 a b \,c^{3}+9 a^{2}}{18 d \,a^{3}}}{\left (d x +c \right )^{2} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2}}+\frac {20 \left (\munderset {\textit {\_R} =\RootOf \left (a^{11} d^{3} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{11} d^{4} \textit {\_R}^{3}-3 b^{2} d \right ) x -4 a^{11} c \,d^{3} \textit {\_R}^{3}-a^{4} b d \textit {\_R} -3 b^{2} c \right )\right )}{27}\) | \(254\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1024 vs.
\(2 (176) = 352\).
time = 0.42, size = 1024, normalized size = 4.68 \begin {gather*} -\frac {60 \, b^{2} d^{6} x^{6} + 360 \, b^{2} c d^{5} x^{5} + 900 \, b^{2} c^{2} d^{4} x^{4} + 60 \, b^{2} c^{6} + 48 \, {\left (25 \, b^{2} c^{3} + 2 \, a b\right )} d^{3} x^{3} + 96 \, a b c^{3} + 36 \, {\left (25 \, b^{2} c^{4} + 8 \, a b c\right )} d^{2} x^{2} + 72 \, {\left (5 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d x - 40 \, \sqrt {3} {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (a d x + a c\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 20 \, {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + {\left (a b d x + a b c\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 40 \, {\left (b^{2} d^{8} x^{8} + 8 \, b^{2} c d^{7} x^{7} + 28 \, b^{2} c^{2} d^{6} x^{6} + 2 \, {\left (28 \, b^{2} c^{3} + a b\right )} d^{5} x^{5} + b^{2} c^{8} + 10 \, {\left (7 \, b^{2} c^{4} + a b c\right )} d^{4} x^{4} + 2 \, a b c^{5} + 4 \, {\left (14 \, b^{2} c^{5} + 5 \, a b c^{2}\right )} d^{3} x^{3} + {\left (28 \, b^{2} c^{6} + 20 \, a b c^{3} + a^{2}\right )} d^{2} x^{2} + a^{2} c^{2} + 2 \, {\left (4 \, b^{2} c^{7} + 5 \, a b c^{4} + a^{2} c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b d x + b c - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 27 \, a^{2}}{54 \, {\left (a^{3} b^{2} d^{9} x^{8} + 8 \, a^{3} b^{2} c d^{8} x^{7} + 28 \, a^{3} b^{2} c^{2} d^{7} x^{6} + 2 \, {\left (28 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{6} x^{5} + 10 \, {\left (7 \, a^{3} b^{2} c^{4} + a^{4} b c\right )} d^{5} x^{4} + 4 \, {\left (14 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{4} x^{3} + {\left (28 \, a^{3} b^{2} c^{6} + 20 \, a^{4} b c^{3} + a^{5}\right )} d^{3} x^{2} + 2 \, {\left (4 \, a^{3} b^{2} c^{7} + 5 \, a^{4} b c^{4} + a^{5} c\right )} d^{2} x + {\left (a^{3} b^{2} c^{8} + 2 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 435 vs.
\(2 (199) = 398\).
time = 3.06, size = 435, normalized size = 1.99 \begin {gather*} \frac {- 9 a^{2} - 32 a b c^{3} - 20 b^{2} c^{6} - 300 b^{2} c^{2} d^{4} x^{4} - 120 b^{2} c d^{5} x^{5} - 20 b^{2} d^{6} x^{6} + x^{3} \left (- 32 a b d^{3} - 400 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 96 a b c d^{2} - 300 b^{2} c^{4} d^{2}\right ) + x \left (- 96 a b c^{2} d - 120 b^{2} c^{5} d\right )}{18 a^{5} c^{2} d + 36 a^{4} b c^{5} d + 18 a^{3} b^{2} c^{8} d + 504 a^{3} b^{2} c^{2} d^{7} x^{6} + 144 a^{3} b^{2} c d^{8} x^{7} + 18 a^{3} b^{2} d^{9} x^{8} + x^{5} \cdot \left (36 a^{4} b d^{6} + 1008 a^{3} b^{2} c^{3} d^{6}\right ) + x^{4} \cdot \left (180 a^{4} b c d^{5} + 1260 a^{3} b^{2} c^{4} d^{5}\right ) + x^{3} \cdot \left (360 a^{4} b c^{2} d^{4} + 1008 a^{3} b^{2} c^{5} d^{4}\right ) + x^{2} \cdot \left (18 a^{5} d^{3} + 360 a^{4} b c^{3} d^{3} + 504 a^{3} b^{2} c^{6} d^{3}\right ) + x \left (36 a^{5} c d^{2} + 180 a^{4} b c^{4} d^{2} + 144 a^{3} b^{2} c^{7} d^{2}\right )} + \frac {\operatorname {RootSum} {\left (19683 t^{3} a^{11} + 8000 b^{2}, \left ( t \mapsto t \log {\left (x + \frac {- 27 t a^{4} + 20 b c}{20 b d} \right )} \right )\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs.
\(2 (176) = 352\).
time = 1.32, size = 365, normalized size = 1.67 \begin {gather*} \frac {10 \, {\left (2 \, \sqrt {3} \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | -b d x - b c + \left (-a b^{2}\right )^{\frac {1}{3}} \right |}\right )\right )}}{27 \, a^{3}} - \frac {20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 400 \, b^{2} c^{3} d^{3} x^{3} + 300 \, b^{2} c^{4} d^{2} x^{2} + 120 \, b^{2} c^{5} d x + 20 \, b^{2} c^{6} + 32 \, a b d^{3} x^{3} + 96 \, a b c d^{2} x^{2} + 96 \, a b c^{2} d x + 32 \, a b c^{3} + 9 \, a^{2}}{18 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a d x + a c\right )}^{2} a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.69, size = 520, normalized size = 2.37 \begin {gather*} \frac {20\,b^{2/3}\,\ln \left (a^6\,b^{1/3}\,c-{\left (-a\right )}^{19/3}+a^6\,b^{1/3}\,d\,x\right )}{27\,{\left (-a\right )}^{11/3}\,d}-\frac {\frac {9\,a^2+32\,a\,b\,c^3+20\,b^2\,c^6}{18\,a^3\,d}+\frac {2\,x^2\,\left (25\,d\,b^2\,c^4+8\,a\,d\,b\,c\right )}{3\,a^3}+\frac {4\,x\,\left (5\,b^2\,c^5+4\,a\,b\,c^2\right )}{3\,a^3}+\frac {8\,x^3\,\left (25\,b^2\,c^3\,d^2+2\,a\,b\,d^2\right )}{9\,a^3}+\frac {10\,b^2\,d^5\,x^6}{9\,a^3}+\frac {50\,b^2\,c^2\,d^3\,x^4}{3\,a^3}+\frac {20\,b^2\,c\,d^4\,x^5}{3\,a^3}}{x^5\,\left (56\,b^2\,c^3\,d^5+2\,a\,b\,d^5\right )+x^4\,\left (70\,b^2\,c^4\,d^4+10\,a\,b\,c\,d^4\right )+x\,\left (2\,d\,a^2\,c+10\,d\,a\,b\,c^4+8\,d\,b^2\,c^7\right )+x^2\,\left (a^2\,d^2+20\,a\,b\,c^3\,d^2+28\,b^2\,c^6\,d^2\right )+a^2\,c^2+b^2\,c^8+x^3\,\left (56\,b^2\,c^5\,d^3+20\,a\,b\,c^2\,d^3\right )+b^2\,d^8\,x^8+2\,a\,b\,c^5+8\,b^2\,c\,d^7\,x^7+28\,b^2\,c^2\,d^6\,x^6}+\frac {20\,b^{2/3}\,\ln \left (4860\,a^6\,b^3\,c\,d^5+4860\,a^6\,b^3\,d^6\,x-4860\,{\left (-a\right )}^{19/3}\,b^{8/3}\,d^5\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,{\left (-a\right )}^{11/3}\,d}-\frac {20\,b^{2/3}\,\ln \left (4860\,a^6\,b^3\,c\,d^5+4860\,a^6\,b^3\,d^6\,x+4860\,{\left (-a\right )}^{19/3}\,b^{8/3}\,d^5\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{27\,{\left (-a\right )}^{11/3}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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