Optimal. Leaf size=237 \[ -\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d e^3}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d e^3}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} d e^3}-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \]
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Rubi [A] time = 0.462997, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{20 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{11/3} d e^3}+\frac{10 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{11/3} d e^3}+\frac{20 b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} d e^3}-\frac{10}{9 a^3 d e^3 (c+d x)^2}+\frac{4}{9 a^2 d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/((c*e + d*e*x)^3*(a + b*(c + d*x)^3)^3),x]
[Out]
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Rubi in Sympy [A] time = 53.3033, size = 226, normalized size = 0.95 \[ \frac{1}{6 a d e^{3} \left (a + b \left (c + d x\right )^{3}\right )^{2} \left (c + d x\right )^{2}} + \frac{4}{9 a^{2} d e^{3} \left (a + b \left (c + d x\right )^{3}\right ) \left (c + d x\right )^{2}} - \frac{10}{9 a^{3} d e^{3} \left (c + d x\right )^{2}} - \frac{20 b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{27 a^{\frac{11}{3}} d e^{3}} + \frac{10 b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{27 a^{\frac{11}{3}} d e^{3}} + \frac{20 \sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{11}{3}} d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*e*x+c*e)**3/(a+b*(d*x+c)**3)**3,x)
[Out]
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Mathematica [A] time = 0.26636, size = 195, normalized size = 0.82 \[ \frac{20 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{9 a^{5/3} b (c+d x)}{\left (a+b (c+d x)^3\right )^2}-\frac{33 a^{2/3} b (c+d x)}{a+b (c+d x)^3}-\frac{27 a^{2/3}}{(c+d x)^2}-40 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-40 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{54 a^{11/3} d e^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*e + d*e*x)^3*(a + b*(c + d*x)^3)^3),x]
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Maple [C] time = 0.014, size = 446, normalized size = 1.9 \[ -{\frac{11\,{b}^{2}{d}^{3}{x}^{4}}{18\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{22\,c{d}^{2}{b}^{2}{x}^{3}}{9\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{11\,{b}^{2}{c}^{2}d{x}^{2}}{3\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{22\,{b}^{2}x{c}^{3}}{9\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{7\,bx}{9\,{e}^{3}{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{11\,{b}^{2}{c}^{4}}{18\,{e}^{3}{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{7\,bc}{9\,{e}^{3}{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{20}{27\,{e}^{3}{a}^{3}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}-{\frac{1}{2\,{e}^{3}{a}^{3}d \left ( dx+c \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*e*x+c*e)^3/(a+b*(d*x+c)^3)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{20 \, b^{2} d^{6} x^{6} + 120 \, b^{2} c d^{5} x^{5} + 300 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{6} + 16 \,{\left (25 \, b^{2} c^{3} + 2 \, a b\right )} d^{3} x^{3} + 32 \, a b c^{3} + 12 \,{\left (25 \, b^{2} c^{4} + 8 \, a b c\right )} d^{2} x^{2} + 24 \,{\left (5 \, b^{2} c^{5} + 4 \, a b c^{2}\right )} d x + 9 \, a^{2}}{18 \,{\left (a^{3} b^{2} d^{9} e^{3} x^{8} + 8 \, a^{3} b^{2} c d^{8} e^{3} x^{7} + 28 \, a^{3} b^{2} c^{2} d^{7} e^{3} x^{6} + 2 \,{\left (28 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{6} e^{3} x^{5} + 10 \,{\left (7 \, a^{3} b^{2} c^{4} + a^{4} b c\right )} d^{5} e^{3} x^{4} + 4 \,{\left (14 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{4} e^{3} x^{3} +{\left (28 \, a^{3} b^{2} c^{6} + 20 \, a^{4} b c^{3} + a^{5}\right )} d^{3} e^{3} x^{2} + 2 \,{\left (4 \, a^{3} b^{2} c^{7} + 5 \, a^{4} b c^{4} + a^{5} c\right )} d^{2} e^{3} x +{\left (a^{3} b^{2} c^{8} + 2 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d e^{3}\right )}} - \frac{20 \, b \int \frac{1}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{9 \, a^{3} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)^3*(d*e*x + c*e)^3),x, algorithm="maxima")
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Fricas [A] time = 0.340866, size = 1447, normalized size = 6.11 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)^3*(d*e*x + c*e)^3),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*e*x+c*e)**3/(a+b*(d*x+c)**3)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{3}{\left (d e x + c e\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)^3*(d*e*x + c*e)^3),x, algorithm="giac")
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