Optimal. Leaf size=158 \[ \frac {\sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3 (a+b x) (d+e x)^4}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \frac {\sqrt {a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{3 e^3 (a+b x) (d+e x)^3}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{4 e^3 (a+b x) (d+e x)^4}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \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 {\left (a b+b^2 x\right ) (A+B 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) (-B d+A e)}{e^2 (d+e x)^5}+\frac {b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^4}+\frac {b^2 B}{e^2 (d+e x)^3}\right ) \, dx}{a b+b^2 x}\\ &=-\frac {(b d-a e) (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}+\frac {(2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^3 (a+b x) (d+e x)^3}-\frac {b B \sqrt {a^2+2 a b x+b^2 x^2}}{2 e^3 (a+b x) (d+e x)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 80, normalized size = 0.51 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a e (3 A e+B (d+4 e x))+b \left (A e (d+4 e x)+B \left (d^2+4 d e x+6 e^2 x^2\right )\right )\right )}{12 e^3 (a+b x) (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 181.13, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 102, normalized size = 0.65 \begin {gather*} -\frac {6 \, B b e^{2} x^{2} + B b d^{2} + 3 \, A a e^{2} + {\left (B a + A b\right )} d e + 4 \, {\left (B b d e + {\left (B a + A b\right )} e^{2}\right )} x}{12 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 116, normalized size = 0.73 \begin {gather*} -\frac {{\left (6 \, B b x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, B b d x e \mathrm {sgn}\left (b x + a\right ) + B b d^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a x e^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, A b x e^{2} \mathrm {sgn}\left (b x + a\right ) + B a d e \mathrm {sgn}\left (b x + a\right ) + A b d e \mathrm {sgn}\left (b x + a\right ) + 3 \, A a e^{2} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{12 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 86, normalized size = 0.54 \begin {gather*} -\frac {\left (6 B b \,e^{2} x^{2}+4 A b \,e^{2} x +4 B a \,e^{2} x +4 B b d e x +3 A a \,e^{2}+A b d e +B a d e +B b \,d^{2}\right ) \sqrt {\left (b x +a \right )^{2}}}{12 \left (e x +d \right )^{4} \left (b x +a \right ) e^{3}} \end {gather*}
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]
________________________________________________________________________________________
mupad [B] time = 2.15, size = 85, normalized size = 0.54 \begin {gather*} -\frac {\sqrt {{\left (a+b\,x\right )}^2}\,\left (3\,A\,a\,e^2+B\,b\,d^2+4\,A\,b\,e^2\,x+4\,B\,a\,e^2\,x+6\,B\,b\,e^2\,x^2+A\,b\,d\,e+B\,a\,d\,e+4\,B\,b\,d\,e\,x\right )}{12\,e^3\,\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]
________________________________________________________________________________________
sympy [A] time = 2.62, size = 117, normalized size = 0.74 \begin {gather*} \frac {- 3 A a e^{2} - A b d e - B a d e - B b d^{2} - 6 B b e^{2} x^{2} + x \left (- 4 A b e^{2} - 4 B a e^{2} - 4 B b d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________