Optimal. Leaf size=158 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{5 e^3 (a+b x) (d+e x)^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{6 e^3 (a+b x) (d+e x)^6}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.095612, antiderivative size = 158, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (-a B e-A b e+2 b B d)}{5 e^3 (a+b x) (d+e x)^5}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (B d-A e)}{6 e^3 (a+b x) (d+e x)^6}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 770
Rule 77
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{a^2+2 a b x+b^2 x^2}}{(d+e x)^7} \, 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)^7} \, 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)^7}+\frac{b (-2 b B d+A b e+a B e)}{e^2 (d+e x)^6}+\frac{b^2 B}{e^2 (d+e x)^5}\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}}{6 e^3 (a+b x) (d+e x)^6}+\frac{(2 b B d-A b e-a B e) \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^3 (a+b x) (d+e x)^5}-\frac{b B \sqrt{a^2+2 a b x+b^2 x^2}}{4 e^3 (a+b x) (d+e x)^4}\\ \end{align*}
Mathematica [A] time = 0.0428492, size = 82, normalized size = 0.52 \[ -\frac{\sqrt{(a+b x)^2} \left (2 a e (5 A e+B (d+6 e x))+b \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )\right )}{60 e^3 (a+b x) (d+e x)^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 88, normalized size = 0.6 \begin{align*} -{\frac{15\,B{x}^{2}b{e}^{2}+12\,Axb{e}^{2}+12\,aB{e}^{2}x+6\,Bxbde+10\,aA{e}^{2}+2\,Abde+2\,aBde+Bb{d}^{2}}{60\,{e}^{3} \left ( ex+d \right ) ^{6} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.56273, size = 273, normalized size = 1.73 \begin{align*} -\frac{15 \, B b e^{2} x^{2} + B b d^{2} + 10 \, A a e^{2} + 2 \,{\left (B a + A b\right )} d e + 6 \,{\left (B b d e + 2 \,{\left (B a + A b\right )} e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.0227, size = 144, normalized size = 0.91 \begin{align*} - \frac{10 A a e^{2} + 2 A b d e + 2 B a d e + B b d^{2} + 15 B b e^{2} x^{2} + x \left (12 A b e^{2} + 12 B a e^{2} + 6 B b d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11721, size = 159, normalized size = 1.01 \begin{align*} -\frac{{\left (15 \, B b x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 6 \, B b d x e \mathrm{sgn}\left (b x + a\right ) + B b d^{2} \mathrm{sgn}\left (b x + a\right ) + 12 \, B a x e^{2} \mathrm{sgn}\left (b x + a\right ) + 12 \, A b x e^{2} \mathrm{sgn}\left (b x + a\right ) + 2 \, B a d e \mathrm{sgn}\left (b x + a\right ) + 2 \, A b d e \mathrm{sgn}\left (b x + a\right ) + 10 \, A a e^{2} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]