3.2028 \(\int \frac{(a+b x) (d+e x)}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=67 \[ \frac{e (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{d+e x}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-((d + e*x)/(b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + (e*(a + b*x)*Log[a + b*x])/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)

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Rubi [A]  time = 0.0293024, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {768, 608, 31} \[ \frac{e (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{d+e x}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

-((d + e*x)/(b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) + (e*(a + b*x)*Log[a + b*x])/(b^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)

Rule 768

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]

Rule 608

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{(a+b x) (d+e x)}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=-\frac{d+e x}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e \int \frac{1}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx}{b}\\ &=-\frac{d+e x}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (e \left (a b+b^2 x\right )\right ) \int \frac{1}{a b+b^2 x} \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{d+e x}{b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e (a+b x) \log (a+b x)}{b^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0167334, size = 36, normalized size = 0.54 \[ \frac{e (a+b x) \log (a+b x)+a e-b d}{b^2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(d + e*x))/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(-(b*d) + a*e + e*(a + b*x)*Log[a + b*x])/(b^2*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.009, size = 48, normalized size = 0.7 \begin{align*}{\frac{ \left ( \ln \left ( bx+a \right ) xbe+\ln \left ( bx+a \right ) ae+ae-bd \right ) \left ( bx+a \right ) ^{2}}{{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

(ln(b*x+a)*x*b*e+ln(b*x+a)*a*e+a*e-b*d)*(b*x+a)^2/b^2/((b*x+a)^2)^(3/2)

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Maxima [B]  time = 1.01217, size = 185, normalized size = 2.76 \begin{align*} \frac{b e \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{3 \, a^{2} b^{3} e}{2 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{2 \, a b^{2} e x}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}} - \frac{b d + a e}{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac{a d}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}}{\left (x + \frac{a}{b}\right )}^{2}} + \frac{{\left (b d + a e\right )} a}{2 \,{\left (b^{2}\right )}^{\frac{3}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

b*e*log(x + a/b)/(b^2)^(3/2) + 3/2*a^2*b^3*e/((b^2)^(7/2)*(x + a/b)^2) + 2*a*b^2*e*x/((b^2)^(5/2)*(x + a/b)^2)
- (b*d + a*e)/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 1/2*a*d/((b^2)^(3/2)*(x + a/b)^2) + 1/2*(b*d + a*e)*a/((b
^2)^(3/2)*b*(x + a/b)^2)

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Fricas [A]  time = 1.71769, size = 80, normalized size = 1.19 \begin{align*} -\frac{b d - a e -{\left (b e x + a e\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

-(b*d - a*e - (b*e*x + a*e)*log(b*x + a))/(b^3*x + a*b^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x\right ) \left (d + e x\right )}{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral((a + b*x)*(d + e*x)/((a + b*x)**2)**(3/2), x)

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Giac [B]  time = 1.21716, size = 154, normalized size = 2.3 \begin{align*} -\frac{e \log \left ({\left | -3 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}^{2} a b - a^{3} b -{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}^{3}{\left | b \right |} - 3 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )} a^{2}{\left | b \right |} \right |}\right )}{3 \, b{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

-1/3*e*log(abs(-3*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2))^2*a*b - a^3*b - (x*abs(b) - sqrt(b^2*x^2 + 2*a*b*
x + a^2))^3*abs(b) - 3*(x*abs(b) - sqrt(b^2*x^2 + 2*a*b*x + a^2))*a^2*abs(b)))/(b*abs(b))