Optimal. Leaf size=228 \[ -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}+\frac {c^3 \log (d+e x)}{e^7} \]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {698} \begin {gather*} -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {c^3 \log (d+e x)}{e^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx &=\int \left (\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^7}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^5}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right )}{e^6 (d+e x)^4}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^6 (d+e x)^3}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^2}+\frac {c^3}{e^6 (d+e x)}\right ) \, dx\\ &=-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 231, normalized size = 1.01 \begin {gather*} \frac {-b^3 e^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )-6 b^2 c e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )-30 b c^2 e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 407, normalized size = 1.79 \begin {gather*} \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \, {\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 260, normalized size = 1.14 \begin {gather*} c^{3} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {{\left (180 \, {\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x + {\left (147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} e^{\left (-1\right )}\right )} e^{\left (-6\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 387, normalized size = 1.70 \begin {gather*} \frac {b^{3} d^{3}}{6 \left (e x +d \right )^{6} e^{4}}-\frac {b^{2} c \,d^{4}}{2 \left (e x +d \right )^{6} e^{5}}+\frac {b \,c^{2} d^{5}}{2 \left (e x +d \right )^{6} e^{6}}-\frac {c^{3} d^{6}}{6 \left (e x +d \right )^{6} e^{7}}-\frac {3 b^{3} d^{2}}{5 \left (e x +d \right )^{5} e^{4}}+\frac {12 b^{2} c \,d^{3}}{5 \left (e x +d \right )^{5} e^{5}}-\frac {3 b \,c^{2} d^{4}}{\left (e x +d \right )^{5} e^{6}}+\frac {6 c^{3} d^{5}}{5 \left (e x +d \right )^{5} e^{7}}+\frac {3 b^{3} d}{4 \left (e x +d \right )^{4} e^{4}}-\frac {9 b^{2} c \,d^{2}}{2 \left (e x +d \right )^{4} e^{5}}+\frac {15 b \,c^{2} d^{3}}{2 \left (e x +d \right )^{4} e^{6}}-\frac {15 c^{3} d^{4}}{4 \left (e x +d \right )^{4} e^{7}}-\frac {b^{3}}{3 \left (e x +d \right )^{3} e^{4}}+\frac {4 b^{2} c d}{\left (e x +d \right )^{3} e^{5}}-\frac {10 b \,c^{2} d^{2}}{\left (e x +d \right )^{3} e^{6}}+\frac {20 c^{3} d^{3}}{3 \left (e x +d \right )^{3} e^{7}}-\frac {3 b^{2} c}{2 \left (e x +d \right )^{2} e^{5}}+\frac {15 b \,c^{2} d}{2 \left (e x +d \right )^{2} e^{6}}-\frac {15 c^{3} d^{2}}{2 \left (e x +d \right )^{2} e^{7}}-\frac {3 b \,c^{2}}{\left (e x +d \right ) e^{6}}+\frac {6 c^{3} d}{\left (e x +d \right ) e^{7}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.53, size = 331, normalized size = 1.45 \begin {gather*} \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {c^{3} \log \left (e x + d\right )}{e^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 268, normalized size = 1.18 \begin {gather*} \frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {x^5\,\left (3\,b\,c^2\,e^6-6\,c^3\,d\,e^5\right )+x^4\,\left (\frac {3\,b^2\,c\,e^6}{2}+\frac {15\,b\,c^2\,d\,e^5}{2}-\frac {45\,c^3\,d^2\,e^4}{2}\right )+x\,\left (\frac {b^3\,d^2\,e^4}{10}+\frac {3\,b^2\,c\,d^3\,e^3}{5}+3\,b\,c^2\,d^4\,e^2-\frac {137\,c^3\,d^5\,e}{10}\right )+x^2\,\left (\frac {b^3\,d\,e^5}{4}+\frac {3\,b^2\,c\,d^2\,e^4}{2}+\frac {15\,b\,c^2\,d^3\,e^3}{2}-\frac {125\,c^3\,d^4\,e^2}{4}\right )+x^3\,\left (\frac {b^3\,e^6}{3}+2\,b^2\,c\,d\,e^5+10\,b\,c^2\,d^2\,e^4-\frac {110\,c^3\,d^3\,e^3}{3}\right )-\frac {49\,c^3\,d^6}{20}+\frac {b^3\,d^3\,e^3}{60}+\frac {b^2\,c\,d^4\,e^2}{10}+\frac {b\,c^2\,d^5\,e}{2}}{e^7\,{\left (d+e\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 30.82, size = 343, normalized size = 1.50 \begin {gather*} \frac {c^{3} \log {\left (d + e x \right )}}{e^{7}} + \frac {- b^{3} d^{3} e^{3} - 6 b^{2} c d^{4} e^{2} - 30 b c^{2} d^{5} e + 147 c^{3} d^{6} + x^{5} \left (- 180 b c^{2} e^{6} + 360 c^{3} d e^{5}\right ) + x^{4} \left (- 90 b^{2} c e^{6} - 450 b c^{2} d e^{5} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 20 b^{3} e^{6} - 120 b^{2} c d e^{5} - 600 b c^{2} d^{2} e^{4} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 15 b^{3} d e^{5} - 90 b^{2} c d^{2} e^{4} - 450 b c^{2} d^{3} e^{3} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 6 b^{3} d^{2} e^{4} - 36 b^{2} c d^{3} e^{3} - 180 b c^{2} d^{4} e^{2} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________