3.198 \(\int \frac{x^6}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=244 \[ -\frac{a^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 a^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{20 a^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a x (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(20*a^3)/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - a^6/(4*b^7*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^5)
/(b^7*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*a^4)/(2*b^7*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(5*a*x*(a + b*x))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (x^2*(a + b*x))/(2*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (15*a^2*(a + b*x)*Log[a + b*x])/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.117451, antiderivative size = 244, 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} \[ -\frac{a^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 a^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{20 a^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a x (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(20*a^3)/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - a^6/(4*b^7*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*a^5)
/(b^7*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (15*a^4)/(2*b^7*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
(5*a*x*(a + b*x))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (x^2*(a + b*x))/(2*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (15*a^2*(a + b*x)*Log[a + b*x])/(b^7*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^6}{\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^6}{\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{5 a}{b^{11}}+\frac{x}{b^{10}}+\frac{a^6}{b^{11} (a+b x)^5}-\frac{6 a^5}{b^{11} (a+b x)^4}+\frac{15 a^4}{b^{11} (a+b x)^3}-\frac{20 a^3}{b^{11} (a+b x)^2}+\frac{15 a^2}{b^{11} (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{20 a^3}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a^6}{4 b^7 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{2 a^5}{b^7 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{15 a^4}{2 b^7 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 a x (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{x^2 (a+b x)}{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{15 a^2 (a+b x) \log (a+b x)}{b^7 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0342297, size = 106, normalized size = 0.43 \[ \frac{132 a^4 b^2 x^2-32 a^3 b^3 x^3-68 a^2 b^4 x^4+168 a^5 b x+60 a^2 (a+b x)^4 \log (a+b x)+57 a^6-12 a b^5 x^5+2 b^6 x^6}{4 b^7 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(57*a^6 + 168*a^5*b*x + 132*a^4*b^2*x^2 - 32*a^3*b^3*x^3 - 68*a^2*b^4*x^4 - 12*a*b^5*x^5 + 2*b^6*x^6 + 60*a^2*
(a + b*x)^4*Log[a + b*x])/(4*b^7*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.264, size = 158, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,{b}^{6}{x}^{6}+60\,\ln \left ( bx+a \right ){x}^{4}{a}^{2}{b}^{4}-12\,{x}^{5}a{b}^{5}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{3}{b}^{3}-68\,{a}^{2}{x}^{4}{b}^{4}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{4}{b}^{2}-32\,{a}^{3}{x}^{3}{b}^{3}+240\,\ln \left ( bx+a \right ) x{a}^{5}b+132\,{a}^{4}{x}^{2}{b}^{2}+60\,\ln \left ( bx+a \right ){a}^{6}+168\,{a}^{5}xb+57\,{a}^{6} \right ) \left ( bx+a \right ) }{4\,{b}^{7}} \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^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/4*(2*b^6*x^6+60*ln(b*x+a)*x^4*a^2*b^4-12*x^5*a*b^5+240*ln(b*x+a)*x^3*a^3*b^3-68*a^2*x^4*b^4+360*ln(b*x+a)*x^
2*a^4*b^2-32*a^3*x^3*b^3+240*ln(b*x+a)*x*a^5*b+132*a^4*x^2*b^2+60*ln(b*x+a)*a^6+168*a^5*x*b+57*a^6)*(b*x+a)/b^
7/((b*x+a)^2)^(5/2)

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Maxima [A]  time = 1.29544, size = 170, normalized size = 0.7 \begin{align*} \frac{2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6}}{4 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} + \frac{15 \, a^{2} \log \left (b x + a\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/4*(2*b^6*x^6 - 12*a*b^5*x^5 - 68*a^2*b^4*x^4 - 32*a^3*b^3*x^3 + 132*a^4*b^2*x^2 + 168*a^5*b*x + 57*a^6)/(b^1
1*x^4 + 4*a*b^10*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7) + 15*a^2*log(b*x + a)/b^7

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Fricas [A]  time = 1.56303, size = 344, normalized size = 1.41 \begin{align*} \frac{2 \, b^{6} x^{6} - 12 \, a b^{5} x^{5} - 68 \, a^{2} b^{4} x^{4} - 32 \, a^{3} b^{3} x^{3} + 132 \, a^{4} b^{2} x^{2} + 168 \, a^{5} b x + 57 \, a^{6} + 60 \,{\left (a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{3} + 6 \, a^{4} b^{2} x^{2} + 4 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{4 \,{\left (b^{11} x^{4} + 4 \, a b^{10} x^{3} + 6 \, a^{2} b^{9} x^{2} + 4 \, a^{3} b^{8} x + a^{4} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/4*(2*b^6*x^6 - 12*a*b^5*x^5 - 68*a^2*b^4*x^4 - 32*a^3*b^3*x^3 + 132*a^4*b^2*x^2 + 168*a^5*b*x + 57*a^6 + 60*
(a^2*b^4*x^4 + 4*a^3*b^3*x^3 + 6*a^4*b^2*x^2 + 4*a^5*b*x + a^6)*log(b*x + a))/(b^11*x^4 + 4*a*b^10*x^3 + 6*a^2
*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\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**6/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(x**6/((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^6/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x