3.2030 \(\int \frac{a+b x}{(d+e x) (a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=120 \[ -\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}+\frac{e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.0839477, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {770, 21, 44} \[ -\frac{1}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}+\frac{e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

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]

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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

Mathematica [A]  time = 0.0383337, size = 57, normalized size = 0.48 \[ \frac{e (a+b x) \log (d+e x)-e (a+b x) \log (a+b x)+a e-b d}{\sqrt{(a+b x)^2} (b d-a e)^2} \]

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

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Maple [A]  time = 0.013, size = 76, normalized size = 0.6 \begin{align*}{\frac{ \left ( \ln \left ( ex+d \right ) xbe-\ln \left ( bx+a \right ) xbe+\ln \left ( ex+d \right ) ae-\ln \left ( bx+a \right ) ae+ae-bd \right ) \left ( bx+a \right ) ^{2}}{ \left ( ae-bd \right ) ^{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(e*x+d)*x*b*e-ln(b*x+a)*x*b*e+ln(e*x+d)*a*e-ln(b*x+a)*a*e+a*e-b*d)*(b*x+a)^2/(a*e-b*d)^2/((b*x+a)^2)^(3/2)

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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]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.69395, size = 200, normalized size = 1.67 \begin{align*} -\frac{b d - a e +{\left (b e x + a e\right )} \log \left (b x + a\right ) -{\left (b e x + a e\right )} \log \left (e x + d\right )}{a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x} \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*e*x + a*e)*log(e*x + d))/(a*b^2*d^2 - 2*a^2*b*d*e + a^3*e^2 + (b
^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b x}{\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 = 1.18918, size = 274, normalized size = 2.28 \begin{align*} -\frac{a e \log \left ({\left | b + \frac{a}{x} \right |}\right )}{a b^{2} d^{2} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 2 \, a^{2} b d e \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{3} e^{2} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{d e \log \left ({\left | \frac{d}{x} + e \right |}\right )}{b^{2} d^{3} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 2 \, a b d^{2} e \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{2} d e^{2} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{b^{2} d - a b e}{{\left (b d - a e\right )}^{2} a{\left (b + \frac{a}{x}\right )} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\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]

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