Optimal. Leaf size=92 \[ \frac {(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {660, 45}
\begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 660
Rubi steps
\begin {align*} \int \frac {\sqrt {a^2+2 a b x+b^2 x^2}}{(d+e x)^5} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {a b+b^2 x}{(d+e x)^5} \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b (b d-a e)}{e (d+e x)^5}+\frac {b^2}{e (d+e x)^4}\right ) \, dx}{a b+b^2 x}\\ &=\frac {(b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^2 (a+b x) (d+e x)^4}-\frac {b \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 45, normalized size = 0.49 \begin {gather*} -\frac {\sqrt {(a+b x)^2} (3 a e+b (d+4 e x))}{12 e^2 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.46, size = 32, normalized size = 0.35
method | result | size |
default | \(-\frac {\mathrm {csgn}\left (b x +a \right ) \left (4 b e x +3 a e +b d \right )}{12 e^{2} \left (e x +d \right )^{4}}\) | \(32\) |
gosper | \(-\frac {\left (4 b e x +3 a e +b d \right ) \sqrt {\left (b x +a \right )^{2}}}{12 e^{2} \left (e x +d \right )^{4} \left (b x +a \right )}\) | \(42\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {b x}{3 e}-\frac {3 a e +b d}{12 e^{2}}\right )}{\left (b x +a \right ) \left (e x +d \right )^{4}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.66, size = 58, normalized size = 0.63 \begin {gather*} -\frac {b d + {\left (4 \, b x + 3 \, a\right )} e}{12 \, {\left (x^{4} e^{6} + 4 \, d x^{3} e^{5} + 6 \, d^{2} x^{2} e^{4} + 4 \, d^{3} x e^{3} + d^{4} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.23, size = 65, normalized size = 0.71 \begin {gather*} \frac {- 3 a e - b d - 4 b e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.68, size = 45, normalized size = 0.49 \begin {gather*} -\frac {{\left (4 \, b x e \mathrm {sgn}\left (b x + a\right ) + b d \mathrm {sgn}\left (b x + a\right ) + 3 \, a e \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-2\right )}}{12 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.59, size = 41, normalized size = 0.45 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (3\,a\,e+b\,d+4\,b\,e\,x\right )}{12\,e^2\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________