### 3.1603 $$\int \frac{1}{(d+e x)^3 (a^2+2 a b x+b^2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=276 $\frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}$

[Out]

(3*b^2*e)/((b*d - a*e)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - b^2/(2*(b*d - a*e)^3*(a + b*x)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + (e^2*(a + b*x))/(2*(b*d - a*e)^3*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b*e^2*(a + b*x))/
((b*d - a*e)^4*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (6*b^2*e^2*(a + b*x)*Log[a + b*x])/((b*d - a*e)^5*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) - (6*b^2*e^2*(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 = 0.151705, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {646, 44} $\frac{3 b e^2 (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x) (b d-a e)^4}+\frac{e^2 (a+b x)}{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^2 (b d-a e)^3}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{6 b^2 e^2 (a+b x) \log (d+e x)}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}-\frac{b^2}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}+\frac{3 b^2 e}{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

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

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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{1}{(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{1}{\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 (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac{1}{(b d-a e)^3 (a+b x)^3}-\frac{3 e}{(b d-a e)^4 (a+b x)^2}+\frac{6 e^2}{(b d-a e)^5 (a+b x)}-\frac{e^3}{b^3 (b d-a e)^3 (d+e x)^3}-\frac{3 e^3}{b^2 (b d-a e)^4 (d+e x)^2}-\frac{6 e^3}{b (b d-a e)^5 (d+e x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{3 b^2 e}{(b d-a e)^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2}{2 (b d-a e)^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^2 (a+b x)}{2 (b d-a e)^3 (d+e x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b e^2 (a+b x)}{(b d-a e)^4 (d+e x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{6 b^2 e^2 (a+b x) \log (a+b x)}{(b d-a e)^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{6 b^2 e^2 (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 = 0.0980738, size = 163, normalized size = 0.59 $\frac{(a+b x) \left (-12 b^2 e^2 (a+b x)^2 \log (d+e x)+6 b^2 e (a+b x) (b d-a e)+b^2 \left (-(b d-a e)^2\right )+12 b^2 e^2 (a+b x)^2 \log (a+b x)+\frac{6 b e^2 (a+b x)^2 (b d-a e)}{d+e x}+\frac{e^2 (a+b x)^2 (b d-a e)^2}{(d+e x)^2}\right )}{2 \left ((a+b x)^2\right )^{3/2} (b d-a e)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)^3*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

((a + b*x)*(-(b^2*(b*d - a*e)^2) + 6*b^2*e*(b*d - a*e)*(a + b*x) + (e^2*(b*d - a*e)^2*(a + b*x)^2)/(d + e*x)^2
+ (6*b*e^2*(b*d - a*e)*(a + b*x)^2)/(d + e*x) + 12*b^2*e^2*(a + b*x)^2*Log[a + b*x] - 12*b^2*e^2*(a + b*x)^2*
Log[d + e*x]))/(2*(b*d - a*e)^5*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.212, size = 508, normalized size = 1.8 \begin{align*}{\frac{ \left ( -24\,\ln \left ( bx+a \right ) xa{b}^{3}{d}^{2}{e}^{2}+24\,\ln \left ( ex+d \right ){x}^{3}a{b}^{3}{e}^{4}-8\,a{b}^{3}{d}^{3}e-48\,\ln \left ( bx+a \right ){x}^{2}a{b}^{3}d{e}^{3}-{a}^{4}{e}^{4}+{b}^{4}{d}^{4}-24\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}{e}^{4}-24\,\ln \left ( bx+a \right ){x}^{3}{b}^{4}d{e}^{3}+12\,\ln \left ( ex+d \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+12\,\ln \left ( ex+d \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+12\,{x}^{3}a{b}^{3}{e}^{4}+48\,\ln \left ( ex+d \right ){x}^{2}a{b}^{3}d{e}^{3}+24\,\ln \left ( ex+d \right ) x{a}^{2}{b}^{2}d{e}^{3}+24\,\ln \left ( ex+d \right ) xa{b}^{3}{d}^{2}{e}^{2}+12\,\ln \left ( ex+d \right ){x}^{4}{b}^{4}{e}^{4}-12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}{e}^{4}-12\,{x}^{3}{b}^{4}d{e}^{3}+18\,{x}^{2}{a}^{2}{b}^{2}{e}^{4}-18\,{x}^{2}{b}^{4}{d}^{2}{e}^{2}+4\,x{a}^{3}b{e}^{4}-4\,x{b}^{4}{d}^{3}e-12\,\ln \left ( bx+a \right ){a}^{2}{b}^{2}{d}^{2}{e}^{2}+24\,x{a}^{2}{b}^{2}d{e}^{3}-24\,xa{b}^{3}{d}^{2}{e}^{2}-24\,\ln \left ( bx+a \right ) x{a}^{2}{b}^{2}d{e}^{3}-12\,\ln \left ( bx+a \right ){x}^{2}{b}^{4}{d}^{2}{e}^{2}-12\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}{e}^{4}+24\,\ln \left ( ex+d \right ){x}^{3}{b}^{4}d{e}^{3}+8\,{a}^{3}bd{e}^{3} \right ) \left ( bx+a \right ) }{2\, \left ( ex+d \right ) ^{2} \left ( ae-bd \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/2*(-24*ln(b*x+a)*x*a*b^3*d^2*e^2+24*ln(e*x+d)*x^3*a*b^3*e^4-8*a*b^3*d^3*e-48*ln(b*x+a)*x^2*a*b^3*d*e^3-a^4*e
^4+b^4*d^4-24*ln(b*x+a)*x^3*a*b^3*e^4-24*ln(b*x+a)*x^3*b^4*d*e^3+12*ln(e*x+d)*x^2*a^2*b^2*e^4+12*ln(e*x+d)*x^2
*b^4*d^2*e^2+12*ln(e*x+d)*a^2*b^2*d^2*e^2+12*x^3*a*b^3*e^4+48*ln(e*x+d)*x^2*a*b^3*d*e^3+24*ln(e*x+d)*x*a^2*b^2
*d*e^3+24*ln(e*x+d)*x*a*b^3*d^2*e^2+12*ln(e*x+d)*x^4*b^4*e^4-12*ln(b*x+a)*x^4*b^4*e^4-12*x^3*b^4*d*e^3+18*x^2*
a^2*b^2*e^4-18*x^2*b^4*d^2*e^2+4*x*a^3*b*e^4-4*x*b^4*d^3*e-12*ln(b*x+a)*a^2*b^2*d^2*e^2+24*x*a^2*b^2*d*e^3-24*
x*a*b^3*d^2*e^2-24*ln(b*x+a)*x*a^2*b^2*d*e^3-12*ln(b*x+a)*x^2*b^4*d^2*e^2-12*ln(b*x+a)*x^2*a^2*b^2*e^4+24*ln(e
*x+d)*x^3*b^4*d*e^3+8*a^3*b*d*e^3)*(b*x+a)/(e*x+d)^2/(a*e-b*d)^5/((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(1/(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 = 1.75502, size = 1488, normalized size = 5.39 \begin{align*} -\frac{b^{4} d^{4} - 8 \, a b^{3} d^{3} e + 8 \, a^{3} b d e^{3} - a^{4} e^{4} - 12 \,{\left (b^{4} d e^{3} - a b^{3} e^{4}\right )} x^{3} - 18 \,{\left (b^{4} d^{2} e^{2} - a^{2} b^{2} e^{4}\right )} x^{2} - 4 \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} - 6 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x - 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} e^{4} x^{4} + a^{2} b^{2} d^{2} e^{2} + 2 \,{\left (b^{4} d e^{3} + a b^{3} e^{4}\right )} x^{3} +{\left (b^{4} d^{2} e^{2} + 4 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \,{\left (a b^{3} d^{2} e^{2} + a^{2} b^{2} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \,{\left (a^{2} b^{5} d^{7} - 5 \, a^{3} b^{4} d^{6} e + 10 \, a^{4} b^{3} d^{5} e^{2} - 10 \, a^{5} b^{2} d^{4} e^{3} + 5 \, a^{6} b d^{3} e^{4} - a^{7} d^{2} e^{5} +{\left (b^{7} d^{5} e^{2} - 5 \, a b^{6} d^{4} e^{3} + 10 \, a^{2} b^{5} d^{3} e^{4} - 10 \, a^{3} b^{4} d^{2} e^{5} + 5 \, a^{4} b^{3} d e^{6} - a^{5} b^{2} e^{7}\right )} x^{4} + 2 \,{\left (b^{7} d^{6} e - 4 \, a b^{6} d^{5} e^{2} + 5 \, a^{2} b^{5} d^{4} e^{3} - 5 \, a^{4} b^{3} d^{2} e^{5} + 4 \, a^{5} b^{2} d e^{6} - a^{6} b e^{7}\right )} x^{3} +{\left (b^{7} d^{7} - a b^{6} d^{6} e - 9 \, a^{2} b^{5} d^{5} e^{2} + 25 \, a^{3} b^{4} d^{4} e^{3} - 25 \, a^{4} b^{3} d^{3} e^{4} + 9 \, a^{5} b^{2} d^{2} e^{5} + a^{6} b d e^{6} - a^{7} e^{7}\right )} x^{2} + 2 \,{\left (a b^{6} d^{7} - 4 \, a^{2} b^{5} d^{6} e + 5 \, a^{3} b^{4} d^{5} e^{2} - 5 \, a^{5} b^{2} d^{3} e^{4} + 4 \, a^{6} b d^{2} e^{5} - a^{7} d e^{6}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right )^{3} \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(1/(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Integral(1/((d + e*x)**3*((a + b*x)**2)**(3/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x