3.2032 \(\int \frac{a+b x}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=223 \[ -\frac{b^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{2 b e (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}-\frac{e (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac{3 b^2 e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{3 b^2 e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

[Out]

-(b^2/((b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (e*(a + b*x))/(2*(b*d - a*e)^2*(d + e*x)^2*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (2*b*e*(a + b*x))/((b*d - a*e)^3*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b^2*e*(a +
b*x)*Log[a + b*x])/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2*e*(a + b*x)*Log[d + e*x])/((b*d - a
*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.147181, antiderivative size = 223, 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} \[ -\frac{b^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{2 b e (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^3}-\frac{e (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^2}-\frac{3 b^2 e (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}+\frac{3 b^2 e (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2/((b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])) - (e*(a + b*x))/(2*(b*d - a*e)^2*(d + e*x)^2*Sqrt[a^2 + 2
*a*b*x + b^2*x^2]) - (2*b*e*(a + b*x))/((b*d - a*e)^3*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (3*b^2*e*(a +
b*x)*Log[a + b*x])/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b^2*e*(a + b*x)*Log[d + e*x])/((b*d - a
*e)^4*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)^3 \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)^3} \, 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)^3} \, 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^3}{(b d-a e)^3 (a+b x)^2}-\frac{3 b^3 e}{(b d-a e)^4 (a+b x)}+\frac{e^2}{(b d-a e)^2 (d+e x)^3}+\frac{2 b e^2}{(b d-a e)^3 (d+e x)^2}+\frac{3 b^2 e^2}{(b d-a e)^4 (d+e x)}\right ) \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{b^2}{(b d-a e)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (a+b x)}{2 (b d-a e)^2 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b e (a+b x)}{(b d-a e)^3 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b^2 e (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2 e (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.10261, size = 135, normalized size = 0.61 \[ \frac{-(b d-a e) \left (-a^2 e^2+a b e (5 d+3 e x)+b^2 \left (2 d^2+9 d e x+6 e^2 x^2\right )\right )-6 b^2 e (a+b x) (d+e x)^2 \log (a+b x)+6 b^2 e (a+b x) (d+e x)^2 \log (d+e x)}{2 \sqrt{(a+b x)^2} (d+e x)^2 (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*d - a*e)*(-(a^2*e^2) + a*b*e*(5*d + 3*e*x) + b^2*(2*d^2 + 9*d*e*x + 6*e^2*x^2))) - 6*b^2*e*(a + b*x)*(d
+ e*x)^2*Log[a + b*x] + 6*b^2*e*(a + b*x)*(d + e*x)^2*Log[d + e*x])/(2*(b*d - a*e)^4*Sqrt[(a + b*x)^2]*(d + e*
x)^2)

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Maple [A]  time = 0.018, size = 332, normalized size = 1.5 \begin{align*}{\frac{ \left ( 6\,\ln \left ( ex+d \right ){x}^{3}{b}^{3}{e}^{3}-6\,\ln \left ( bx+a \right ){x}^{3}{b}^{3}{e}^{3}+6\,\ln \left ( ex+d \right ){x}^{2}a{b}^{2}{e}^{3}+12\,\ln \left ( ex+d \right ){x}^{2}{b}^{3}d{e}^{2}-6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}{e}^{3}-12\,\ln \left ( bx+a \right ){x}^{2}{b}^{3}d{e}^{2}+12\,\ln \left ( ex+d \right ) xa{b}^{2}d{e}^{2}+6\,\ln \left ( ex+d \right ) x{b}^{3}{d}^{2}e-12\,\ln \left ( bx+a \right ) xa{b}^{2}d{e}^{2}-6\,\ln \left ( bx+a \right ) x{b}^{3}{d}^{2}e+6\,{x}^{2}a{b}^{2}{e}^{3}-6\,{x}^{2}{b}^{3}d{e}^{2}+6\,\ln \left ( ex+d \right ) a{b}^{2}{d}^{2}e-6\,\ln \left ( bx+a \right ) a{b}^{2}{d}^{2}e+3\,x{a}^{2}b{e}^{3}+6\,xa{b}^{2}d{e}^{2}-9\,x{b}^{3}{d}^{2}e-{e}^{3}{a}^{3}+6\,d{e}^{2}{a}^{2}b-3\,a{d}^{2}e{b}^{2}-2\,{d}^{3}{b}^{3} \right ) \left ( bx+a \right ) ^{2}}{2\, \left ( ex+d \right ) ^{2} \left ( ae-bd \right ) ^{4}} \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)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/2*(6*ln(e*x+d)*x^3*b^3*e^3-6*ln(b*x+a)*x^3*b^3*e^3+6*ln(e*x+d)*x^2*a*b^2*e^3+12*ln(e*x+d)*x^2*b^3*d*e^2-6*ln
(b*x+a)*x^2*a*b^2*e^3-12*ln(b*x+a)*x^2*b^3*d*e^2+12*ln(e*x+d)*x*a*b^2*d*e^2+6*ln(e*x+d)*x*b^3*d^2*e-12*ln(b*x+
a)*x*a*b^2*d*e^2-6*ln(b*x+a)*x*b^3*d^2*e+6*x^2*a*b^2*e^3-6*x^2*b^3*d*e^2+6*ln(e*x+d)*a*b^2*d^2*e-6*ln(b*x+a)*a
*b^2*d^2*e+3*x*a^2*b*e^3+6*x*a*b^2*d*e^2-9*x*b^3*d^2*e-e^3*a^3+6*d*e^2*a^2*b-3*a*d^2*e*b^2-2*d^3*b^3)*(b*x+a)^
2/(e*x+d)^2/(a*e-b*d)^4/((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)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.96984, size = 991, normalized size = 4.44 \begin{align*} -\frac{2 \, b^{3} d^{3} + 3 \, a b^{2} d^{2} e - 6 \, a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} + 3 \,{\left (3 \, b^{3} d^{2} e - 2 \, a b^{2} d e^{2} - a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e +{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} +{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + a b^{2} d^{2} e +{\left (2 \, b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} +{\left (b^{3} d^{2} e + 2 \, a b^{2} d e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a b^{4} d^{6} - 4 \, a^{2} b^{3} d^{5} e + 6 \, a^{3} b^{2} d^{4} e^{2} - 4 \, a^{4} b d^{3} e^{3} + a^{5} d^{2} e^{4} +{\left (b^{5} d^{4} e^{2} - 4 \, a b^{4} d^{3} e^{3} + 6 \, a^{2} b^{3} d^{2} e^{4} - 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{3} +{\left (2 \, b^{5} d^{5} e - 7 \, a b^{4} d^{4} e^{2} + 8 \, a^{2} b^{3} d^{3} e^{3} - 2 \, a^{3} b^{2} d^{2} e^{4} - 2 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{2} +{\left (b^{5} d^{6} - 2 \, a b^{4} d^{5} e - 2 \, a^{2} b^{3} d^{4} e^{2} + 8 \, a^{3} b^{2} d^{3} e^{3} - 7 \, a^{4} b d^{2} e^{4} + 2 \, a^{5} d e^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b^3*d^3 + 3*a*b^2*d^2*e - 6*a^2*b*d*e^2 + a^3*e^3 + 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 + 3*(3*b^3*d^2*e - 2
*a*b^2*d*e^2 - a^2*b*e^3)*x + 6*(b^3*e^3*x^3 + a*b^2*d^2*e + (2*b^3*d*e^2 + a*b^2*e^3)*x^2 + (b^3*d^2*e + 2*a*
b^2*d*e^2)*x)*log(b*x + a) - 6*(b^3*e^3*x^3 + a*b^2*d^2*e + (2*b^3*d*e^2 + a*b^2*e^3)*x^2 + (b^3*d^2*e + 2*a*b
^2*d*e^2)*x)*log(e*x + d))/(a*b^4*d^6 - 4*a^2*b^3*d^5*e + 6*a^3*b^2*d^4*e^2 - 4*a^4*b*d^3*e^3 + a^5*d^2*e^4 +
(b^5*d^4*e^2 - 4*a*b^4*d^3*e^3 + 6*a^2*b^3*d^2*e^4 - 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^3 + (2*b^5*d^5*e - 7*a*b^4
*d^4*e^2 + 8*a^2*b^3*d^3*e^3 - 2*a^3*b^2*d^2*e^4 - 2*a^4*b*d*e^5 + a^5*e^6)*x^2 + (b^5*d^6 - 2*a*b^4*d^5*e - 2
*a^2*b^3*d^4*e^2 + 8*a^3*b^2*d^3*e^3 - 7*a^4*b*d^2*e^4 + 2*a^5*d*e^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 3.83192, size = 622, normalized size = 2.79 \begin{align*} -\frac{3 \, a b^{2} e \log \left ({\left | b + \frac{a}{x} \right |}\right )}{a b^{4} d^{4} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 4 \, a^{2} b^{3} d^{3} e \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + 6 \, a^{3} b^{2} d^{2} e^{2} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 4 \, a^{4} b d e^{3} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{5} e^{4} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{3 \, b^{2} d e \log \left ({\left | \frac{d}{x} + e \right |}\right )}{b^{4} d^{5} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 4 \, a b^{3} d^{4} e \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + 6 \, a^{2} b^{2} d^{3} e^{2} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) - 4 \, a^{3} b d^{2} e^{3} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right ) + a^{4} d e^{4} \mathrm{sgn}\left (\frac{b}{x} + \frac{a}{x^{2}}\right )} + \frac{2 \, b^{4} d^{3} e^{2} + 3 \, a b^{3} d^{2} e^{3} - 6 \, a^{2} b^{2} d e^{4} + a^{3} b e^{5} + \frac{4 \, b^{4} d^{4} e + 2 \, a b^{3} d^{3} e^{2} - 3 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}}{x} + \frac{2 \,{\left (b^{4} d^{5} - a b^{3} d^{4} e + 3 \, a^{2} b^{2} d^{3} e^{2} - 4 \, a^{3} b d^{2} e^{3} + a^{4} d e^{4}\right )}}{x^{2}}}{2 \,{\left (b d - a e\right )}^{4} a{\left (b + \frac{a}{x}\right )} d^{2}{\left (\frac{d}{x} + e\right )}^{2} \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)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

-3*a*b^2*e*log(abs(b + a/x))/(a*b^4*d^4*sgn(b/x + a/x^2) - 4*a^2*b^3*d^3*e*sgn(b/x + a/x^2) + 6*a^3*b^2*d^2*e^
2*sgn(b/x + a/x^2) - 4*a^4*b*d*e^3*sgn(b/x + a/x^2) + a^5*e^4*sgn(b/x + a/x^2)) + 3*b^2*d*e*log(abs(d/x + e))/
(b^4*d^5*sgn(b/x + a/x^2) - 4*a*b^3*d^4*e*sgn(b/x + a/x^2) + 6*a^2*b^2*d^3*e^2*sgn(b/x + a/x^2) - 4*a^3*b*d^2*
e^3*sgn(b/x + a/x^2) + a^4*d*e^4*sgn(b/x + a/x^2)) + 1/2*(2*b^4*d^3*e^2 + 3*a*b^3*d^2*e^3 - 6*a^2*b^2*d*e^4 +
a^3*b*e^5 + (4*b^4*d^4*e + 2*a*b^3*d^3*e^2 - 3*a^2*b^2*d^2*e^3 - 4*a^3*b*d*e^4 + a^4*e^5)/x + 2*(b^4*d^5 - a*b
^3*d^4*e + 3*a^2*b^2*d^3*e^2 - 4*a^3*b*d^2*e^3 + a^4*d*e^4)/x^2)/((b*d - a*e)^4*a*(b + a/x)*d^2*(d/x + e)^2*sg
n(b/x + a/x^2))