Optimal. Leaf size=136 \[ \frac {243 d^3 (c+d x)^{4/3}}{1820 (a+b x)^{4/3} (b c-a d)^4}-\frac {81 d^2 (c+d x)^{4/3}}{455 (a+b x)^{7/3} (b c-a d)^3}+\frac {27 d (c+d x)^{4/3}}{130 (a+b x)^{10/3} (b c-a d)^2}-\frac {3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {45, 37} \begin {gather*} \frac {243 d^3 (c+d x)^{4/3}}{1820 (a+b x)^{4/3} (b c-a d)^4}-\frac {81 d^2 (c+d x)^{4/3}}{455 (a+b x)^{7/3} (b c-a d)^3}+\frac {27 d (c+d x)^{4/3}}{130 (a+b x)^{10/3} (b c-a d)^2}-\frac {3 (c+d x)^{4/3}}{13 (a+b x)^{13/3} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{16/3}} \, dx &=-\frac {3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}-\frac {(9 d) \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{13/3}} \, dx}{13 (b c-a d)}\\ &=-\frac {3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}+\frac {27 d (c+d x)^{4/3}}{130 (b c-a d)^2 (a+b x)^{10/3}}+\frac {\left (27 d^2\right ) \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{10/3}} \, dx}{65 (b c-a d)^2}\\ &=-\frac {3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}+\frac {27 d (c+d x)^{4/3}}{130 (b c-a d)^2 (a+b x)^{10/3}}-\frac {81 d^2 (c+d x)^{4/3}}{455 (b c-a d)^3 (a+b x)^{7/3}}-\frac {\left (81 d^3\right ) \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{7/3}} \, dx}{455 (b c-a d)^3}\\ &=-\frac {3 (c+d x)^{4/3}}{13 (b c-a d) (a+b x)^{13/3}}+\frac {27 d (c+d x)^{4/3}}{130 (b c-a d)^2 (a+b x)^{10/3}}-\frac {81 d^2 (c+d x)^{4/3}}{455 (b c-a d)^3 (a+b x)^{7/3}}+\frac {243 d^3 (c+d x)^{4/3}}{1820 (b c-a d)^4 (a+b x)^{4/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 118, normalized size = 0.87 \begin {gather*} \frac {3 (c+d x)^{4/3} \left (455 a^3 d^3+195 a^2 b d^2 (3 d x-4 c)+39 a b^2 d \left (14 c^2-12 c d x+9 d^2 x^2\right )+b^3 \left (-140 c^3+126 c^2 d x-108 c d^2 x^2+81 d^3 x^3\right )\right )}{1820 (a+b x)^{13/3} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.13, size = 95, normalized size = 0.70 \begin {gather*} -\frac {3 (c+d x)^{4/3} \left (\frac {140 b^3 (c+d x)^3}{(a+b x)^3}-\frac {546 b^2 d (c+d x)^2}{(a+b x)^2}+\frac {780 b d^2 (c+d x)}{a+b x}-455 d^3\right )}{1820 (a+b x)^{4/3} (b c-a d)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.11, size = 533, normalized size = 3.92 \begin {gather*} \frac {3 \, {\left (81 \, b^{3} d^{4} x^{4} - 140 \, b^{3} c^{4} + 546 \, a b^{2} c^{3} d - 780 \, a^{2} b c^{2} d^{2} + 455 \, a^{3} c d^{3} - 27 \, {\left (b^{3} c d^{3} - 13 \, a b^{2} d^{4}\right )} x^{3} + 9 \, {\left (2 \, b^{3} c^{2} d^{2} - 13 \, a b^{2} c d^{3} + 65 \, a^{2} b d^{4}\right )} x^{2} - {\left (14 \, b^{3} c^{3} d - 78 \, a b^{2} c^{2} d^{2} + 195 \, a^{2} b c d^{3} - 455 \, a^{3} d^{4}\right )} x\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{1820 \, {\left (a^{5} b^{4} c^{4} - 4 \, a^{6} b^{3} c^{3} d + 6 \, a^{7} b^{2} c^{2} d^{2} - 4 \, a^{8} b c d^{3} + a^{9} d^{4} + {\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{5} + 5 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{4} + 10 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{3} + 10 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x^{2} + 5 \, {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {16}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 171, normalized size = 1.26 \begin {gather*} \frac {3 \left (d x +c \right )^{\frac {4}{3}} \left (81 b^{3} d^{3} x^{3}+351 a \,b^{2} d^{3} x^{2}-108 b^{3} c \,d^{2} x^{2}+585 a^{2} b \,d^{3} x -468 a \,b^{2} c \,d^{2} x +126 b^{3} c^{2} d x +455 a^{3} d^{3}-780 a^{2} b c \,d^{2}+546 a \,b^{2} c^{2} d -140 b^{3} c^{3}\right )}{1820 \left (b x +a \right )^{\frac {13}{3}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (d x + c\right )}^{\frac {1}{3}}}{{\left (b x + a\right )}^{\frac {16}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.15, size = 293, normalized size = 2.15 \begin {gather*} \frac {{\left (c+d\,x\right )}^{1/3}\,\left (\frac {243\,d^4\,x^4}{1820\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {-1365\,a^3\,c\,d^3+2340\,a^2\,b\,c^2\,d^2-1638\,a\,b^2\,c^3\,d+420\,b^3\,c^4}{1820\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {x\,\left (1365\,a^3\,d^4-585\,a^2\,b\,c\,d^3+234\,a\,b^2\,c^2\,d^2-42\,b^3\,c^3\,d\right )}{1820\,b^4\,{\left (a\,d-b\,c\right )}^4}+\frac {81\,d^3\,x^3\,\left (13\,a\,d-b\,c\right )}{1820\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {27\,d^2\,x^2\,\left (65\,a^2\,d^2-13\,a\,b\,c\,d+2\,b^2\,c^2\right )}{1820\,b^3\,{\left (a\,d-b\,c\right )}^4}\right )}{x^4\,{\left (a+b\,x\right )}^{1/3}+\frac {a^4\,{\left (a+b\,x\right )}^{1/3}}{b^4}+\frac {6\,a^2\,x^2\,{\left (a+b\,x\right )}^{1/3}}{b^2}+\frac {4\,a\,x^3\,{\left (a+b\,x\right )}^{1/3}}{b}+\frac {4\,a^3\,x\,{\left (a+b\,x\right )}^{1/3}}{b^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________