Optimal. Leaf size=219 \[ -\frac{7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d}+\frac{14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}+\frac{14 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} d}-\frac{14}{9 a^3 d (c+d x)}+\frac{7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.436296, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ -\frac{7 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{27 a^{10/3} d}+\frac{14 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{27 a^{10/3} d}+\frac{14 \sqrt [3]{b} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} d}-\frac{14}{9 a^3 d (c+d x)}+\frac{7}{18 a^2 d (c+d x) \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d (c+d x) \left (a+b (c+d x)^3\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/((c + d*x)^2*(a + b*(c + d*x)^3)^3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 50.6071, size = 201, normalized size = 0.92 \[ \frac{1}{6 a d \left (a + b \left (c + d x\right )^{3}\right )^{2} \left (c + d x\right )} + \frac{7}{18 a^{2} d \left (a + b \left (c + d x\right )^{3}\right ) \left (c + d x\right )} - \frac{14}{9 a^{3} d \left (c + d x\right )} + \frac{14 \sqrt [3]{b} \log{\left (\sqrt [3]{a} + \sqrt [3]{b} \left (c + d x\right ) \right )}}{27 a^{\frac{10}{3}} d} - \frac{7 \sqrt [3]{b} \log{\left (a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{b} \left (- c - d x\right ) + b^{\frac{2}{3}} \left (c + d x\right )^{2} \right )}}{27 a^{\frac{10}{3}} d} + \frac{14 \sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \sqrt [3]{b} \left (- \frac{2 c}{3} - \frac{2 d x}{3}\right )\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{10}{3}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*x+c)**2/(a+b*(d*x+c)**3)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.257518, size = 196, normalized size = 0.89 \[ \frac{-14 \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac{9 a^{4/3} b (c+d x)^2}{\left (a+b (c+d x)^3\right )^2}-\frac{30 \sqrt [3]{a} b (c+d x)^2}{a+b (c+d x)^3}+28 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-28 \sqrt{3} \sqrt [3]{b} \tan ^{-1}\left (\frac{2 \sqrt [3]{b} (c+d x)-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{54 \sqrt [3]{a}}{c+d x}}{54 a^{10/3} d} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c + d*x)^2*(a + b*(c + d*x)^3)^3),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.035, size = 524, normalized size = 2.4 \[ -{\frac{5\,{b}^{2}{d}^{4}{x}^{5}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{25\,{b}^{2}c{d}^{3}{x}^{4}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{50\,{b}^{2}{c}^{2}{d}^{2}{x}^{3}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{50\,{b}^{2}{x}^{2}{c}^{3}d}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{13\,bd{x}^{2}}{18\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{25\,{b}^{2}x{c}^{4}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{13\,bxc}{9\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}-{\frac{5\,{b}^{2}{c}^{5}}{9\,{a}^{3} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{13\,b{c}^{2}}{18\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{14}{27\,{a}^{3}d}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}b{d}^{3}+3\,{{\it \_Z}}^{2}bc{d}^{2}+3\,{\it \_Z}\,b{c}^{2}d+b{c}^{3}+a \right ) }{\frac{ \left ({\it \_R}\,d+c \right ) \ln \left ( x-{\it \_R} \right ) }{{d}^{2}{{\it \_R}}^{2}+2\,cd{\it \_R}+{c}^{2}}}}-{\frac{1}{{a}^{3}d \left ( dx+c \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*x+c)^2/(a+b*(d*x+c)^3)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\frac{28 \, b^{2} d^{6} x^{6} + 168 \, b^{2} c d^{5} x^{5} + 420 \, b^{2} c^{2} d^{4} x^{4} + 28 \, b^{2} c^{6} + 7 \,{\left (80 \, b^{2} c^{3} + 7 \, a b\right )} d^{3} x^{3} + 49 \, a b c^{3} + 21 \,{\left (20 \, b^{2} c^{4} + 7 \, a b c\right )} d^{2} x^{2} + 21 \,{\left (8 \, b^{2} c^{5} + 7 \, a b c^{2}\right )} d x + 18 \, a^{2}}{18 \,{\left (a^{3} b^{2} d^{8} x^{7} + 7 \, a^{3} b^{2} c d^{7} x^{6} + 21 \, a^{3} b^{2} c^{2} d^{6} x^{5} +{\left (35 \, a^{3} b^{2} c^{3} + 2 \, a^{4} b\right )} d^{5} x^{4} +{\left (35 \, a^{3} b^{2} c^{4} + 8 \, a^{4} b c\right )} d^{4} x^{3} + 3 \,{\left (7 \, a^{3} b^{2} c^{5} + 4 \, a^{4} b c^{2}\right )} d^{3} x^{2} +{\left (7 \, a^{3} b^{2} c^{6} + 8 \, a^{4} b c^{3} + a^{5}\right )} d^{2} x +{\left (a^{3} b^{2} c^{7} + 2 \, a^{4} b c^{4} + a^{5} c\right )} d\right )}} - \frac{14 \, b \int \frac{d x + c}{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}\,{d x}}{9 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)^3*(d*x + c)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.305301, size = 1191, normalized size = 5.44 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)^3*(d*x + c)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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*x+c)**2/(a+b*(d*x+c)**3)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.229755, size = 301, normalized size = 1.37 \[ \frac{14 \, \left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | -\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}} - \frac{1}{{\left (d x + c\right )} d} \right |}\right )}{27 \, a^{3}} - \frac{14 \, \sqrt{3} \left (a^{2} b\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}} - \frac{2}{{\left (d x + c\right )} d}\right )}}{3 \, \left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} d} - \frac{7 \, \left (a^{2} b\right )^{\frac{1}{3}}{\rm ln}\left (\left (\frac{b}{a d^{3}}\right )^{\frac{2}{3}} - \frac{\left (\frac{b}{a d^{3}}\right )^{\frac{1}{3}}}{{\left (d x + c\right )} d} + \frac{1}{{\left (d x + c\right )}^{2} d^{2}}\right )}{27 \, a^{4} d} - \frac{\frac{10 \, b^{2}}{{\left (d x + c\right )} d} + \frac{13 \, a b}{{\left (d x + c\right )}^{4} d}}{18 \, a^{3}{\left (b + \frac{a}{{\left (d x + c\right )}^{3}}\right )}^{2}} - \frac{1}{{\left (d x + c\right )} a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(((d*x + c)^3*b + a)^3*(d*x + c)^2),x, algorithm="giac")
[Out]