3.2018 \(\int \frac{(a+b x) (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=62 \[ \frac{d x (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0301969, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {770, 21} \[ \frac{d x (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e x^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rubi steps

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

Mathematica [A]  time = 0.0094198, size = 28, normalized size = 0.45 \[ \frac{x (a+b x) (2 d+e x)}{2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 25, normalized size = 0.4 \begin{align*}{\frac{x \left ( ex+2\,d \right ) \left ( bx+a \right ) }{2}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)/((b*x+a)^2)^(1/2),x)

[Out]

1/2*x*(e*x+2*d)*(b*x+a)/((b*x+a)^2)^(1/2)

________________________________________________________________________________________

Maxima [B]  time = 0.964697, size = 159, normalized size = 2.56 \begin{align*} \frac{a^{2} b^{3} e \log \left (x + \frac{a}{b}\right )}{{\left (b^{2}\right )}^{\frac{5}{2}}} - \frac{a b^{2} e x}{{\left (b^{2}\right )}^{\frac{3}{2}}} + \frac{b e x^{2}}{2 \, \sqrt{b^{2}}} + a \sqrt{\frac{1}{b^{2}}} d \log \left (x + \frac{a}{b}\right ) - \frac{{\left (b d + a e\right )} a \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right )}{b} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2}}{\left (b d + a e\right )}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.48111, size = 23, normalized size = 0.37 \begin{align*} \frac{1}{2} \, e x^{2} + d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*e*x^2 + d*x

________________________________________________________________________________________

Sympy [A]  time = 0.089606, size = 8, normalized size = 0.13 \begin{align*} d x + \frac{e x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/((b*x+a)**2)**(1/2),x)

[Out]

d*x + e*x**2/2

________________________________________________________________________________________

Giac [A]  time = 1.14817, size = 26, normalized size = 0.42 \begin{align*} \frac{1}{2} \,{\left (x^{2} e + 2 \, d x\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)/((b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

1/2*(x^2*e + 2*d*x)*sgn(b*x + a)