Optimal. Leaf size=137 \[ -\frac {b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac {d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac {2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac {c^2}{2 e^5 (d+e x)^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 137, 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 {b^2 e^2-6 b c d e+6 c^2 d^2}{4 e^5 (d+e x)^4}-\frac {d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}+\frac {2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac {c^2}{2 e^5 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 698
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^2}{(d+e x)^7} \, dx &=\int \left (\frac {d^2 (c d-b e)^2}{e^4 (d+e x)^7}+\frac {2 d (c d-b e) (-2 c d+b e)}{e^4 (d+e x)^6}+\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{e^4 (d+e x)^5}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^4}+\frac {c^2}{e^4 (d+e x)^3}\right ) \, dx\\ &=-\frac {d^2 (c d-b e)^2}{6 e^5 (d+e x)^6}+\frac {2 d (c d-b e) (2 c d-b e)}{5 e^5 (d+e x)^5}-\frac {6 c^2 d^2-6 b c d e+b^2 e^2}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 116, normalized size = 0.85 \begin {gather*} -\frac {b^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 b c e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 c^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 )}{60 e^5 (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 )^2}{(d+e x)^7} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 191, normalized size = 1.39 \begin {gather*} -\frac {30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \, {\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \, {\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \, {\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 132, normalized size = 0.96 \begin {gather*} -\frac {{\left (30 \, c^{2} x^{4} e^{4} + 40 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 12 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 40 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 12 \, b c d^{2} x e^{2} + 2 \, b c d^{3} e + 15 \, b^{2} x^{2} e^{4} + 6 \, b^{2} d x e^{3} + b^{2} d^{2} e^{2}\right )} e^{\left (-5\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 = 0.06, size = 143, normalized size = 1.04 \begin {gather*} -\frac {c^{2}}{2 \left (e x +d \right )^{2} e^{5}}-\frac {\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) d^{2}}{6 \left (e x +d \right )^{6} e^{5}}-\frac {2 \left (b e -2 c d \right ) c}{3 \left (e x +d \right )^{3} e^{5}}+\frac {2 \left (b^{2} e^{2}-3 b c d e +2 c^{2} d^{2}\right ) d}{5 \left (e x +d \right )^{5} e^{5}}-\frac {b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{4 \left (e x +d \right )^{4} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.46, size = 191, normalized size = 1.39 \begin {gather*} -\frac {30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 40 \, {\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \, {\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 6 \, {\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.09, size = 181, normalized size = 1.32 \begin {gather*} -\frac {\frac {x^2\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{4\,e^3}+\frac {c^2\,x^4}{2\,e}+\frac {d^2\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{60\,e^5}+\frac {2\,c\,x^3\,\left (b\,e+c\,d\right )}{3\,e^2}+\frac {d\,x\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2\right )}{10\,e^4}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 4.11, size = 207, normalized size = 1.51 \begin {gather*} \frac {- b^{2} d^{2} e^{2} - 2 b c d^{3} e - 2 c^{2} d^{4} - 30 c^{2} e^{4} x^{4} + x^{3} \left (- 40 b c e^{4} - 40 c^{2} d e^{3}\right ) + x^{2} \left (- 15 b^{2} e^{4} - 30 b c d e^{3} - 30 c^{2} d^{2} e^{2}\right ) + x \left (- 6 b^{2} d e^{3} - 12 b c d^{2} e^{2} - 12 c^{2} d^{3} e\right )}{60 d^{6} e^{5} + 360 d^{5} e^{6} x + 900 d^{4} e^{7} x^{2} + 1200 d^{3} e^{8} x^{3} + 900 d^{2} e^{9} x^{4} + 360 d e^{10} x^{5} + 60 e^{11} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________