Optimal. Leaf size=182 \[ \frac {a+b x}{2 (b d-a e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{(b d-a e)^2 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) \log (a+b x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x) \log (d+e x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 182, 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, 46}
\begin {gather*} \frac {b (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^2}+\frac {a+b x}{2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)}+\frac {b^2 (a+b x) \log (a+b x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {b^2 (a+b x) \log (d+e x)}{\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 46
Rule 660
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {1}{\left (a b+b^2 x\right ) (d+e x)^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (a b+b^2 x\right ) \int \left (\frac {b^2}{(b d-a e)^3 (a+b x)}-\frac {e}{b (b d-a e) (d+e x)^3}-\frac {e}{(b d-a e)^2 (d+e x)^2}-\frac {b e}{(b d-a e)^3 (d+e x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {a+b x}{2 (b d-a e) (d+e x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b (a+b x)}{(b d-a e)^2 (d+e x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {b^2 (a+b x) \log (a+b x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {b^2 (a+b x) \log (d+e x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 97, normalized size = 0.53 \begin {gather*} \frac {(a+b x) \left ((b d-a e) (3 b d-a e+2 b e x)+2 b^2 (d+e x)^2 \log (a+b x)-2 b^2 (d+e x)^2 \log (d+e x)\right )}{2 (b d-a e)^3 \sqrt {(a+b x)^2} (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.56, size = 162, normalized size = 0.89
method | result | size |
default | \(-\frac {\left (b x +a \right ) \left (2 \ln \left (b x +a \right ) b^{2} e^{2} x^{2}-2 \ln \left (e x +d \right ) b^{2} e^{2} x^{2}+4 \ln \left (b x +a \right ) b^{2} d e x -4 \ln \left (e x +d \right ) b^{2} d e x +2 \ln \left (b x +a \right ) b^{2} d^{2}-2 \ln \left (e x +d \right ) b^{2} d^{2}-2 a b \,e^{2} x +2 b^{2} d e x +a^{2} e^{2}-4 a b d e +3 b^{2} d^{2}\right )}{2 \sqrt {\left (b x +a \right )^{2}}\, \left (a e -b d \right )^{3} \left (e x +d \right )^{2}}\) | \(162\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {b e x}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}-\frac {a e -3 b d}{2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}\right )}{\left (b x +a \right ) \left (e x +d \right )^{2}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, b^{2} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(219\) |
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 = 3.84, size = 241, normalized size = 1.32 \begin {gather*} \frac {3 \, b^{2} d^{2} - {\left (2 \, a b x - a^{2}\right )} e^{2} + 2 \, {\left (b^{2} d x - 2 \, a b d\right )} e + 2 \, {\left (b^{2} x^{2} e^{2} + 2 \, b^{2} d x e + b^{2} d^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} x^{2} e^{2} + 2 \, b^{2} d x e + b^{2} d^{2}\right )} \log \left (x e + d\right )}{2 \, {\left (b^{3} d^{5} - a^{3} x^{2} e^{5} + {\left (3 \, a^{2} b d x^{2} - 2 \, a^{3} d x\right )} e^{4} - {\left (3 \, a b^{2} d^{2} x^{2} - 6 \, a^{2} b d^{2} x + a^{3} d^{2}\right )} e^{3} + {\left (b^{3} d^{3} x^{2} - 6 \, a b^{2} d^{3} x + 3 \, a^{2} b d^{3}\right )} e^{2} + {\left (2 \, b^{3} d^{4} x - 3 \, a b^{2} d^{4}\right )} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs.
\(2 (129) = 258\).
time = 0.62, size = 381, normalized size = 2.09 \begin {gather*} \frac {b^{2} \log {\left (x + \frac {- \frac {a^{4} b^{2} e^{4}}{\left (a e - b d\right )^{3}} + \frac {4 a^{3} b^{3} d e^{3}}{\left (a e - b d\right )^{3}} - \frac {6 a^{2} b^{4} d^{2} e^{2}}{\left (a e - b d\right )^{3}} + \frac {4 a b^{5} d^{3} e}{\left (a e - b d\right )^{3}} + a b^{2} e - \frac {b^{6} d^{4}}{\left (a e - b d\right )^{3}} + b^{3} d}{2 b^{3} e} \right )}}{\left (a e - b d\right )^{3}} - \frac {b^{2} \log {\left (x + \frac {\frac {a^{4} b^{2} e^{4}}{\left (a e - b d\right )^{3}} - \frac {4 a^{3} b^{3} d e^{3}}{\left (a e - b d\right )^{3}} + \frac {6 a^{2} b^{4} d^{2} e^{2}}{\left (a e - b d\right )^{3}} - \frac {4 a b^{5} d^{3} e}{\left (a e - b d\right )^{3}} + a b^{2} e + \frac {b^{6} d^{4}}{\left (a e - b d\right )^{3}} + b^{3} d}{2 b^{3} e} \right )}}{\left (a e - b d\right )^{3}} + \frac {- a e + 3 b d + 2 b e x}{2 a^{2} d^{2} e^{2} - 4 a b d^{3} e + 2 b^{2} d^{4} + x^{2} \cdot \left (2 a^{2} e^{4} - 4 a b d e^{3} + 2 b^{2} d^{2} e^{2}\right ) + x \left (4 a^{2} d e^{3} - 8 a b d^{2} e^{2} + 4 b^{2} d^{3} e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.38, size = 174, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, {\left (\frac {2 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} - \frac {2 \, b^{2} e \log \left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} + \frac {3 \, b^{2} d^{2} - 4 \, a b d e + a^{2} e^{2} + 2 \, {\left (b^{2} d e - a b e^{2}\right )} x}{{\left (b d - a e\right )}^{3} {\left (x e + d\right )}^{2}}\right )} \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {{\left (a+b\,x\right )}^2}\,{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________