Optimal. Leaf size=38 \[ -\frac {1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {642, 607} \begin {gather*} -\frac {1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 607
Rule 642
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \, dx &=c \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {1}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 26, normalized size = 0.68 \begin {gather*} -\frac {c (d+e x)}{4 e \left (c (d+e x)^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 1.10, size = 338, normalized size = 8.89 \begin {gather*} \frac {2 c^2 \left (c d^8 e-15 c d^4 e^5 x^4-64 c d^3 e^6 x^5-96 c d^2 e^7 x^6-64 c d e^8 x^7-16 c e^9 x^8\right )+2 c^2 \sqrt {c e^2} \left (d^7-d^6 e x+d^5 e^2 x^2-d^4 e^3 x^3+16 d^3 e^4 x^4+48 d^2 e^5 x^5+48 d e^6 x^6+16 e^7 x^7\right ) \sqrt {c d^2+2 c d e x+c e^2 x^2}}{d^4 e x^4 \sqrt {c d^2+2 c d e x+c e^2 x^2} \left (-8 c^4 d^3 e^5-24 c^4 d^2 e^6 x-24 c^4 d e^7 x^2-8 c^4 e^8 x^3\right )+d^4 e x^4 \sqrt {c e^2} \left (8 c^4 d^4 e^4+32 c^4 d^3 e^5 x+48 c^4 d^2 e^6 x^2+32 c^4 d e^7 x^3+8 c^4 e^8 x^4\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 97, normalized size = 2.55 \begin {gather*} -\frac {\sqrt {c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{4 \, {\left (c^{2} e^{6} x^{5} + 5 \, c^{2} d e^{5} x^{4} + 10 \, c^{2} d^{2} e^{4} x^{3} + 10 \, c^{2} d^{3} e^{3} x^{2} + 5 \, c^{2} d^{4} e^{2} x + c^{2} d^{5} e\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*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 35, normalized size = 0.92 \begin {gather*} -\frac {1}{4 \left (e x +d \right ) \left (c \,e^{2} x^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.36, size = 61, normalized size = 1.61 \begin {gather*} -\frac {1}{4 \, {\left (c^{\frac {3}{2}} e^{5} x^{4} + 4 \, c^{\frac {3}{2}} d e^{4} x^{3} + 6 \, c^{\frac {3}{2}} d^{2} e^{3} x^{2} + 4 \, c^{\frac {3}{2}} d^{3} e^{2} x + c^{\frac {3}{2}} d^{4} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.48, size = 37, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{4\,c^2\,e\,{\left (d+e\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c \left (d + e x\right )^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________