### 3.199 $$\int \frac{x^5}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx$$

Optimal. Leaf size=205 $\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}$

[Out]

(-10*a^2)/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + a^5/(4*b^6*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*a^4
)/(3*b^6*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*a^3)/(b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*a*(a + b*x)*Log[a + b*x])/(b^6*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])

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Rubi [A]  time = 0.0930691, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.083, Rules used = {646, 43} $\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-10*a^2)/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + a^5/(4*b^6*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*a^4
)/(3*b^6*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*a^3)/(b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*a*(a + b*x)*Log[a + b*x])/(b^6*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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^5}{\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{x^5}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \left (\frac{1}{b^{10}}-\frac{a^5}{b^{10} (a+b x)^5}+\frac{5 a^4}{b^{10} (a+b x)^4}-\frac{10 a^3}{b^{10} (a+b x)^3}+\frac{10 a^2}{b^{10} (a+b x)^2}-\frac{5 a}{b^{10} (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{10 a^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{a^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 a^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a (a+b x) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0294583, size = 93, normalized size = 0.45 $\frac{-252 a^3 b^2 x^2-48 a^2 b^3 x^3-248 a^4 b x-77 a^5+48 a b^4 x^4-60 a (a+b x)^4 \log (a+b x)+12 b^5 x^5}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-77*a^5 - 248*a^4*b*x - 252*a^3*b^2*x^2 - 48*a^2*b^3*x^3 + 48*a*b^4*x^4 + 12*b^5*x^5 - 60*a*(a + b*x)^4*Log[a
+ b*x])/(12*b^6*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.221, size = 145, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 60\,\ln \left ( bx+a \right ){x}^{4}a{b}^{4}-12\,{b}^{5}{x}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}-48\,a{b}^{4}{x}^{4}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}+48\,{a}^{2}{b}^{3}{x}^{3}+240\,\ln \left ( bx+a \right ) x{a}^{4}b+252\,{a}^{3}{b}^{2}{x}^{2}+60\,\ln \left ( bx+a \right ){a}^{5}+248\,{a}^{4}bx+77\,{a}^{5} \right ) \left ( bx+a \right ) }{12\,{b}^{6}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(60*ln(b*x+a)*x^4*a*b^4-12*b^5*x^5+240*ln(b*x+a)*x^3*a^2*b^3-48*a*b^4*x^4+360*ln(b*x+a)*x^2*a^3*b^2+48*a
^2*b^3*x^3+240*ln(b*x+a)*x*a^4*b+252*a^3*b^2*x^2+60*ln(b*x+a)*a^5+248*a^4*b*x+77*a^5)*(b*x+a)/b^6/((b*x+a)^2)^
(5/2)

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Maxima [A]  time = 1.25088, size = 153, normalized size = 0.75 \begin{align*} \frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5}}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} - \frac{5 \, a \log \left (b x + a\right )}{b^{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*b^5*x^5 + 48*a*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x - 77*a^5)/(b^10*x^4 + 4*a*b^9
*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6) - 5*a*log(b*x + a)/b^6

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Fricas [A]  time = 1.71691, size = 320, normalized size = 1.56 \begin{align*} \frac{12 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} - 48 \, a^{2} b^{3} x^{3} - 252 \, a^{3} b^{2} x^{2} - 248 \, a^{4} b x - 77 \, a^{5} - 60 \,{\left (a b^{4} x^{4} + 4 \, a^{2} b^{3} x^{3} + 6 \, a^{3} b^{2} x^{2} + 4 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*b^5*x^5 + 48*a*b^4*x^4 - 48*a^2*b^3*x^3 - 252*a^3*b^2*x^2 - 248*a^4*b*x - 77*a^5 - 60*(a*b^4*x^4 + 4*
a^2*b^3*x^3 + 6*a^3*b^2*x^2 + 4*a^4*b*x + a^5)*log(b*x + a))/(b^10*x^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b
^7*x + a^4*b^6)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

sage0*x