3.2.77 \(\int \frac {x^5}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)
Optimal. Leaf size=205 \[ -\frac {10 a^2}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a (a+b x) \log (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {a^5}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 a^4}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 a^3}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.09, antiderivative size = 205, normalized size of antiderivative = 1.00,
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} \begin {gather*} \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}} \end {gather*}
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 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])
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]
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*}
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Mathematica [A] time = 0.03, size = 93, normalized size = 0.45 \begin {gather*} \frac {-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^4-60 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}} \end {gather*}
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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IntegrateAlgebraic [B] time = 2.32, size = 2357, normalized size = 11.50 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[x^5/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
((512*a^8*x)/(3*b^4*Sqrt[b^2]) + (2528*a^7*x^2)/(3*b^3*Sqrt[b^2]) + (2624*a^6*x^3)/(b^2)^(3/2) + (15808*a^5*x^
4)/(3*b*Sqrt[b^2]) + (18688*a^4*x^5)/(3*Sqrt[b^2]) + (3584*a^3*b*x^6)/Sqrt[b^2] + 256*a^2*Sqrt[b^2]*x^7 - (576
*a*b^3*x^8)/Sqrt[b^2] - (128*b^4*x^9)/Sqrt[b^2] - (32*a^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^6 - (416*a^7*x*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])/(3*b^5) - (704*a^6*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^4 - (1920*a^5*x^3*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/b^3 - (10048*a^4*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^2) - (2880*a^3*x^5*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/b - 704*a^2*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2] + 448*a*b*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2] +
128*b^2*x^8*Sqrt[a^2 + 2*a*b*x + b^2*x^2] - (640*a^5*x^4*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/a])/b^2 - (2560*a^4*x^5*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/b - 3840*a^3*x^6*Arc
Tanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a] - 2560*a^2*b*x^7*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 +
2*a*b*x + b^2*x^2])/a] - 640*a*b^2*x^8*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a] + (640*a^4
*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/(b*Sqrt[b^2])
+ (1920*a^3*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2])/a])/Sqr
t[b^2] + (1920*a^2*b*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b*x + b^2*x^2]
)/a])/Sqrt[b^2] + 640*a*Sqrt[b^2]*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*ArcTanh[(-(Sqrt[b^2]*x) + Sqrt[a^2 + 2*a*b
*x + b^2*x^2])/a])/((-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2])^4*(a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])^4) + ((-32*a^9)/(b^5*Sqrt[b^2]) - (512*a^8*x)/(3*b^4*Sqrt[b^2]) - (2528*a^7*x^2)/(3*b^3*Sqrt[b^2]
) - (2624*a^6*x^3)/(b^2)^(3/2) - (4512*a^5*x^4)/(b*Sqrt[b^2]) - (3840*a^4*x^5)/Sqrt[b^2] - (1280*a^3*b*x^6)/Sq
rt[b^2] + (512*a^7*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*b^5) + (672*a^6*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^4
+ (1952*a^5*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^3 + (2560*a^4*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b^2 + (1280*
a^3*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/b + (320*a^5*x^4*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])
/(b*Sqrt[b^2]) + (1280*a^4*x^5*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/Sqrt[b^2] + (1920*a^3*b*
x^6*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/Sqrt[b^2] + 1280*a^2*Sqrt[b^2]*x^7*Log[-a - Sqrt[b^
2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]] + (320*a*b^3*x^8*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/
Sqrt[b^2] - (320*a^4*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/
b^2 - (960*a^3*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/b - 96
0*a^2*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]] - 320*a*b*x^7*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]] + (320*a^5*x^4*Log[a - Sqrt[
b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/(b*Sqrt[b^2]) + (1280*a^4*x^5*Log[a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*
x + b^2*x^2]])/Sqrt[b^2] + (1920*a^3*b*x^6*Log[a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/Sqrt[b^2] + 1
280*a^2*Sqrt[b^2]*x^7*Log[a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]] + (320*a*b^3*x^8*Log[a - Sqrt[b^2]*
x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/Sqrt[b^2] - (320*a^4*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[a - Sqrt[b^2]*x
+ Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/b^2 - (960*a^3*x^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[a - Sqrt[b^2]*x + Sqrt
[a^2 + 2*a*b*x + b^2*x^2]])/b - 960*a^2*x^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a
*b*x + b^2*x^2]] - 320*a*b*x^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*Log[a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^
2]])/((-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2])^4*(a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2])^4
)
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fricas [A] time = 0.42, size = 149, normalized size = 0.73 \begin {gather*} \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 {gather*}
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
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
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maple [A] time = 0.06, size = 145, normalized size = 0.71 \begin {gather*} -\frac {\left (60 a \,b^{4} x^{4} \ln \left (b x +a \right )-12 b^{5} x^{5}+240 a^{2} b^{3} x^{3} \ln \left (b x +a \right )-48 a \,b^{4} x^{4}+360 a^{3} b^{2} x^{2} \ln \left (b x +a \right )+48 a^{2} b^{3} x^{3}+240 a^{4} b x \ln \left (b x +a \right )+252 a^{3} b^{2} x^{2}+60 a^{5} \ln \left (b x +a \right )+248 a^{4} b x +77 a^{5}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{6}} \end {gather*}
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 = 1.55, size = 113, normalized size = 0.55 \begin {gather*} \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 {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
[Out]
int(x^5/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
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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