3.2040 \(\int \frac{a+b x}{(d+e x)^2 (a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=260 \[ -\frac{e^3 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac{3 b e^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{4 b e^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{4 b e^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{b e}{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.185768, antiderivative size = 260, 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} \[ -\frac{e^3 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}-\frac{3 b e^2}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}-\frac{4 b e^3 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{4 b e^3 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}+\frac{b e}{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac{b}{3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-3*b*e^2)/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b/(3*(b*d - a*e)^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (b*e)/((b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(a + b*x))/((b*d - a*e)^4*(d
+ e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (4*b*e^3*(a + b*x)*Log[a + b*x])/((b*d - a*e)^5*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (4*b*e^3*(a + b*x)*Log[d + e*x])/((b*d - a*e)^5*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)^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{a+b x}{\left (a b+b^2 x\right )^5 (d+e x)^2} \, 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)^4 (d+e x)^2} \, 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^2}{(b d-a e)^2 (a+b x)^4}-\frac{2 b^2 e}{(b d-a e)^3 (a+b x)^3}+\frac{3 b^2 e^2}{(b d-a e)^4 (a+b x)^2}-\frac{4 b^2 e^3}{(b d-a e)^5 (a+b x)}+\frac{e^4}{(b d-a e)^4 (d+e x)^2}+\frac{4 b e^4}{(b d-a e)^5 (d+e x)}\right ) \, dx}{b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{3 b e^2}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b}{3 (b d-a e)^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b e}{(b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e^3 (a+b x)}{(b d-a e)^4 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 b e^3 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{4 b e^3 (a+b x) \log (d+e x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.111246, size = 144, normalized size = 0.55 \[ \frac{-9 b e^2 (a+b x)^2 (b d-a e)+\frac{3 e^3 (a+b x)^3 (a e-b d)}{d+e x}+12 b e^3 (a+b x)^3 \log (d+e x)+3 b e (a+b x) (b d-a e)^2-b (b d-a e)^3-12 b e^3 (a+b x)^3 \log (a+b x)}{3 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/((d + e*x)^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-(b*(b*d - a*e)^3) + 3*b*e*(b*d - a*e)^2*(a + b*x) - 9*b*e^2*(b*d - a*e)*(a + b*x)^2 + (3*e^3*(-(b*d) + a*e)*
(a + b*x)^3)/(d + e*x) - 12*b*e^3*(a + b*x)^3*Log[a + b*x] + 12*b*e^3*(a + b*x)^3*Log[d + e*x])/(3*(b*d - a*e)
^5*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.02, size = 484, normalized size = 1.9 \begin{align*} -{\frac{ \left ( -36\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}-36\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}+3\,{a}^{4}{e}^{4}-{b}^{4}{d}^{4}-6\,x{a}^{2}{b}^{2}d{e}^{3}-18\,xa{b}^{3}{d}^{2}{e}^{2}-24\,{x}^{2}a{b}^{3}d{e}^{3}+12\,\ln \left ( ex+d \right ){a}^{3}bd{e}^{3}+36\,\ln \left ( ex+d \right ){x}^{2}a{b}^{3}d{e}^{3}+36\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}-12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}{e}^{4}+12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}-12\,{x}^{3}{b}^{4}d{e}^{3}+30\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-6\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+22\,x{a}^{3}b{e}^{4}+2\,x{b}^{4}{d}^{3}e+12\,{x}^{3}a{b}^{3}{e}^{4}+10\,d{e}^{3}{a}^{3}b+6\,a{b}^{3}{d}^{3}e-18\,{a}^{2}{b}^{2}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ) x{a}^{3}b{e}^{4}+36\,\ln \left ( ex+d \right ){x}^{3}a{b}^{3}{e}^{4}+12\,\ln \left ( ex+d \right ){x}^{3}{b}^{4}d{e}^{3}-12\,\ln \left ( bx+a \right ){a}^{3}bd{e}^{3}-12\,\ln \left ( bx+a \right ) x{a}^{3}b{e}^{4}-36\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{4}-12\,\ln \left ( bx+a \right ){x}^{3}{b}^{4}d{e}^{3}-36\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+36\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4} \right ) \left ( bx+a \right ) ^{2}}{ \left ( 3\,ex+3\,d \right ) \left ( ae-bd \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/3*(-36*ln(b*x+a)*x*a^2*b^2*d*e^3-36*ln(b*x+a)*x^2*a*b^3*d*e^3+3*a^4*e^4-b^4*d^4-6*x*a^2*b^2*d*e^3-18*x*a*b^
3*d^2*e^2-24*x^2*a*b^3*d*e^3+12*ln(e*x+d)*a^3*b*d*e^3+36*ln(e*x+d)*x^2*a*b^3*d*e^3+36*ln(e*x+d)*x*a^2*b^2*d*e^
3-12*ln(b*x+a)*x^4*b^4*e^4+12*ln(e*x+d)*x^4*b^4*e^4-12*x^3*b^4*d*e^3+30*x^2*a^2*b^2*e^4-6*x^2*b^4*d^2*e^2+22*x
*a^3*b*e^4+2*x*b^4*d^3*e+12*x^3*a*b^3*e^4+10*d*e^3*a^3*b+6*a*b^3*d^3*e-18*a^2*b^2*d^2*e^2+12*ln(e*x+d)*x*a^3*b
*e^4+36*ln(e*x+d)*x^3*a*b^3*e^4+12*ln(e*x+d)*x^3*b^4*d*e^3-12*ln(b*x+a)*a^3*b*d*e^3-12*ln(b*x+a)*x*a^3*b*e^4-3
6*ln(b*x+a)*x^3*a*b^3*e^4-12*ln(b*x+a)*x^3*b^4*d*e^3-36*ln(b*x+a)*x^2*a^2*b^2*e^4+36*ln(e*x+d)*x^2*a^2*b^2*e^4
)*(b*x+a)^2/(e*x+d)/(a*e-b*d)^5/((b*x+a)^2)^(5/2)

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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)^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.46795, size = 1501, normalized size = 5.77 \begin{align*} -\frac{b^{4} d^{4} - 6 \, a b^{3} d^{3} e + 18 \, a^{2} b^{2} d^{2} e^{2} - 10 \, a^{3} b d e^{3} - 3 \, a^{4} e^{4} + 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} + 6 \,{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} - 5 \, a^{2} b^{2} e^{4}\right )} x^{2} - 2 \,{\left (b^{4} d^{3} e - 9 \, a b^{3} d^{2} e^{2} - 3 \, a^{2} b^{2} d e^{3} + 11 \, a^{3} b e^{4}\right )} x + 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (b x + a\right ) - 12 \,{\left (b^{4} e^{4} x^{4} + a^{3} b d e^{3} +{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (a^{3} b^{5} d^{6} - 5 \, a^{4} b^{4} d^{5} e + 10 \, a^{5} b^{3} d^{4} e^{2} - 10 \, a^{6} b^{2} d^{3} e^{3} + 5 \, a^{7} b d^{2} e^{4} - a^{8} d e^{5} +{\left (b^{8} d^{5} e - 5 \, a b^{7} d^{4} e^{2} + 10 \, a^{2} b^{6} d^{3} e^{3} - 10 \, a^{3} b^{5} d^{2} e^{4} + 5 \, a^{4} b^{4} d e^{5} - a^{5} b^{3} e^{6}\right )} x^{4} +{\left (b^{8} d^{6} - 2 \, a b^{7} d^{5} e - 5 \, a^{2} b^{6} d^{4} e^{2} + 20 \, a^{3} b^{5} d^{3} e^{3} - 25 \, a^{4} b^{4} d^{2} e^{4} + 14 \, a^{5} b^{3} d e^{5} - 3 \, a^{6} b^{2} e^{6}\right )} x^{3} + 3 \,{\left (a b^{7} d^{6} - 4 \, a^{2} b^{6} d^{5} e + 5 \, a^{3} b^{5} d^{4} e^{2} - 5 \, a^{5} b^{3} d^{2} e^{4} + 4 \, a^{6} b^{2} d e^{5} - a^{7} b e^{6}\right )} x^{2} +{\left (3 \, a^{2} b^{6} d^{6} - 14 \, a^{3} b^{5} d^{5} e + 25 \, a^{4} b^{4} d^{4} e^{2} - 20 \, a^{5} b^{3} d^{3} e^{3} + 5 \, a^{6} b^{2} d^{2} e^{4} + 2 \, a^{7} b d e^{5} - a^{8} e^{6}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/3*(b^4*d^4 - 6*a*b^3*d^3*e + 18*a^2*b^2*d^2*e^2 - 10*a^3*b*d*e^3 - 3*a^4*e^4 + 12*(b^4*d*e^3 - a*b^3*e^4)*x
^3 + 6*(b^4*d^2*e^2 + 4*a*b^3*d*e^3 - 5*a^2*b^2*e^4)*x^2 - 2*(b^4*d^3*e - 9*a*b^3*d^2*e^2 - 3*a^2*b^2*d*e^3 +
11*a^3*b*e^4)*x + 12*(b^4*e^4*x^4 + a^3*b*d*e^3 + (b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4
)*x^2 + (3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*log(b*x + a) - 12*(b^4*e^4*x^4 + a^3*b*d*e^3 + (b^4*d*e^3 + 3*a*b^3*e
^4)*x^3 + 3*(a*b^3*d*e^3 + a^2*b^2*e^4)*x^2 + (3*a^2*b^2*d*e^3 + a^3*b*e^4)*x)*log(e*x + d))/(a^3*b^5*d^6 - 5*
a^4*b^4*d^5*e + 10*a^5*b^3*d^4*e^2 - 10*a^6*b^2*d^3*e^3 + 5*a^7*b*d^2*e^4 - a^8*d*e^5 + (b^8*d^5*e - 5*a*b^7*d
^4*e^2 + 10*a^2*b^6*d^3*e^3 - 10*a^3*b^5*d^2*e^4 + 5*a^4*b^4*d*e^5 - a^5*b^3*e^6)*x^4 + (b^8*d^6 - 2*a*b^7*d^5
*e - 5*a^2*b^6*d^4*e^2 + 20*a^3*b^5*d^3*e^3 - 25*a^4*b^4*d^2*e^4 + 14*a^5*b^3*d*e^5 - 3*a^6*b^2*e^6)*x^3 + 3*(
a*b^7*d^6 - 4*a^2*b^6*d^5*e + 5*a^3*b^5*d^4*e^2 - 5*a^5*b^3*d^2*e^4 + 4*a^6*b^2*d*e^5 - a^7*b*e^6)*x^2 + (3*a^
2*b^6*d^6 - 14*a^3*b^5*d^5*e + 25*a^4*b^4*d^4*e^2 - 20*a^5*b^3*d^3*e^3 + 5*a^6*b^2*d^2*e^4 + 2*a^7*b*d*e^5 - a
^8*e^6)*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)**2/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^2), x)