3.1605 \(\int \frac{(d+e x)^5}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=253 \[ -\frac{10 e^3 (b d-a e)^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e^2 (b d-a e)^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e^4 (a+b x) (b d-a e) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (b d-a e)^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-10*e^3*(b*d - a*e)^2)/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*d - a*e)^5/(4*b^6*(a + b*x)^3*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]) - (5*e*(b*d - a*e)^4)/(3*b^6*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*e^2*(b*d - a*e)^3
)/(b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^5*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e
^4*(b*d - a*e)*(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 = 0.175031, antiderivative size = 253, 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, 43} \[ -\frac{10 e^3 (b d-a e)^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e^2 (b d-a e)^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e^4 (a+b x) (b d-a e) \log (a+b x)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (b d-a e)^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*e^3*(b*d - a*e)^2)/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (b*d - a*e)^5/(4*b^6*(a + b*x)^3*Sqrt[a^2 + 2*a*
b*x + b^2*x^2]) - (5*e*(b*d - a*e)^4)/(3*b^6*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*e^2*(b*d - a*e)^3
)/(b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^5*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e
^4*(b*d - a*e)*(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{(d+e 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{(d+e 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{e^5}{b^{10}}+\frac{(b d-a e)^5}{b^{10} (a+b x)^5}+\frac{5 e (b d-a e)^4}{b^{10} (a+b x)^4}+\frac{10 e^2 (b d-a e)^3}{b^{10} (a+b x)^3}+\frac{10 e^3 (b d-a e)^2}{b^{10} (a+b x)^2}+\frac{5 e^4 (b d-a e)}{b^{10} (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{10 e^3 (b d-a e)^2}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^5}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e (b d-a e)^4}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 e^2 (b d-a e)^3}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^5 x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 e^4 (b d-a e) (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 = 0.129426, size = 242, normalized size = 0.96 \[ \frac{-2 a^2 b^3 e^2 \left (60 d^2 e x+5 d^3-270 d e^2 x^2+24 e^3 x^3\right )-2 a^3 b^2 e^3 \left (15 d^2-220 d e x+126 e^2 x^2\right )+a^4 b e^4 (125 d-248 e x)-77 a^5 e^5+a b^4 e \left (-180 d^2 e^2 x^2-40 d^3 e x-5 d^4+240 d e^3 x^3+48 e^4 x^4\right )-60 e^4 (a+b x)^4 (a e-b d) \log (a+b x)+b^5 \left (-\left (60 d^3 e^2 x^2+120 d^2 e^3 x^3+20 d^4 e x+3 d^5-12 e^5 x^5\right )\right )}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-77*a^5*e^5 + a^4*b*e^4*(125*d - 248*e*x) - 2*a^3*b^2*e^3*(15*d^2 - 220*d*e*x + 126*e^2*x^2) - 2*a^2*b^3*e^2*
(5*d^3 + 60*d^2*e*x - 270*d*e^2*x^2 + 24*e^3*x^3) + a*b^4*e*(-5*d^4 - 40*d^3*e*x - 180*d^2*e^2*x^2 + 240*d*e^3
*x^3 + 48*e^4*x^4) - b^5*(3*d^5 + 20*d^4*e*x + 60*d^3*e^2*x^2 + 120*d^2*e^3*x^3 - 12*e^5*x^5) - 60*e^4*(-(b*d)
+ a*e)*(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 [B]  time = 0.204, size = 449, normalized size = 1.8 \begin{align*} -{\frac{ \left ( 77\,{a}^{5}{e}^{5}+3\,{b}^{5}{d}^{5}-125\,d{e}^{4}{a}^{4}b+20\,x{b}^{5}{d}^{4}e-48\,{x}^{4}a{b}^{4}{e}^{5}+48\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}+120\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+252\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}+60\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+248\,x{a}^{4}b{e}^{5}-240\,\ln \left ( bx+a \right ){x}^{3}a{b}^{4}d{e}^{4}+180\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-540\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}-240\,{x}^{3}a{b}^{4}d{e}^{4}+60\,\ln \left ( bx+a \right ){a}^{5}{e}^{5}+10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+30\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-12\,{x}^{5}{b}^{5}{e}^{5}+40\,xa{b}^{4}{d}^{3}{e}^{2}-440\,x{a}^{3}{b}^{2}d{e}^{4}+120\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}+360\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}{e}^{5}+240\,\ln \left ( bx+a \right ) x{a}^{4}b{e}^{5}-60\,\ln \left ( bx+a \right ){a}^{4}bd{e}^{4}+60\,\ln \left ( bx+a \right ){x}^{4}a{b}^{4}{e}^{5}-60\,\ln \left ( bx+a \right ){x}^{4}{b}^{5}d{e}^{4}+240\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}{e}^{5}-360\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{4}-240\,\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}d{e}^{4}+5\,a{b}^{4}{d}^{4}e \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((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/12*(77*a^5*e^5+3*b^5*d^5-125*d*e^4*a^4*b+20*x*b^5*d^4*e-48*x^4*a*b^4*e^5+48*x^3*a^2*b^3*e^5+120*x^3*b^5*d^2
*e^3+252*x^2*a^3*b^2*e^5+60*x^2*b^5*d^3*e^2+248*x*a^4*b*e^5-240*ln(b*x+a)*x^3*a*b^4*d*e^4+180*x^2*a*b^4*d^2*e^
3-540*x^2*a^2*b^3*d*e^4-240*x^3*a*b^4*d*e^4+60*ln(b*x+a)*a^5*e^5+10*a^2*b^3*d^3*e^2+30*a^3*b^2*d^2*e^3-12*x^5*
b^5*e^5+40*x*a*b^4*d^3*e^2-440*x*a^3*b^2*d*e^4+120*x*a^2*b^3*d^2*e^3+360*ln(b*x+a)*x^2*a^3*b^2*e^5+240*ln(b*x+
a)*x*a^4*b*e^5-60*ln(b*x+a)*a^4*b*d*e^4+60*ln(b*x+a)*x^4*a*b^4*e^5-60*ln(b*x+a)*x^4*b^5*d*e^4+240*ln(b*x+a)*x^
3*a^2*b^3*e^5-360*ln(b*x+a)*x^2*a^2*b^3*d*e^4-240*ln(b*x+a)*x*a^3*b^2*d*e^4+5*a*b^4*d^4*e)*(b*x+a)/b^6/((b*x+a
)^2)^(5/2)

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Maxima [B]  time = 1.18969, size = 664, normalized size = 2.62 \begin{align*} \frac{1}{12} \, e^{5}{\left (\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}}{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}} - \frac{60 \, a \log \left (b x + a\right )}{b^{6}}\right )} + \frac{5}{12} \, d e^{4}{\left (\frac{48 \, a b^{3} x^{3} + 108 \, a^{2} b^{2} x^{2} + 88 \, a^{3} b x + 25 \, a^{4}}{b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}} + \frac{12 \, \log \left (b x + a\right )}{b^{5}}\right )} - \frac{5}{6} \, d^{2} e^{3}{\left (\frac{12 \, x^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{8 \, a^{2}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{4}} + \frac{3 \, a^{3} b}{{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} - \frac{8 \, a^{2}}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} + \frac{6 \, a}{{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{2}} - \frac{6 \, a^{3}}{{\left (b^{2}\right )}^{\frac{5}{2}} b^{3}{\left (x + \frac{a}{b}\right )}^{4}}\right )} - \frac{5}{12} \, d^{4} e{\left (\frac{4}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{3 \, a}{{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}}\right )} - \frac{5}{6} \, d^{3} e^{2}{\left (\frac{3 \, a^{2} b^{2}}{{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x + \frac{a}{b}\right )}^{4}} - \frac{8 \, a b}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x + \frac{a}{b}\right )}^{3}} + \frac{6}{{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{2}}\right )} - \frac{d^{5}}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*e^5*((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) - 60*a*log(b*x + a)/b^6) + 5/12*d*e^4*((48*a*b^3*x^3 + 108*
a^2*b^2*x^2 + 88*a^3*b*x + 25*a^4)/(b^9*x^4 + 4*a*b^8*x^3 + 6*a^2*b^7*x^2 + 4*a^3*b^6*x + a^4*b^5) + 12*log(b*
x + a)/b^5) - 5/6*d^2*e^3*(12*x^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) + 8*a^2/((b^2*x^2 + 2*a*b*x + a^2)^(3/
2)*b^4) + 3*a^3*b/((b^2)^(9/2)*(x + a/b)^4) - 8*a^2/((b^2)^(7/2)*(x + a/b)^3) + 6*a/((b^2)^(5/2)*b*(x + a/b)^2
) - 6*a^3/((b^2)^(5/2)*b^3*(x + a/b)^4)) - 5/12*d^4*e*(4/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 3*a/((b^2)^(5
/2)*b*(x + a/b)^4)) - 5/6*d^3*e^2*(3*a^2*b^2/((b^2)^(9/2)*(x + a/b)^4) - 8*a*b/((b^2)^(7/2)*(x + a/b)^3) + 6/(
(b^2)^(5/2)*(x + a/b)^2)) - 1/4*d^5/((b^2)^(5/2)*(x + a/b)^4)

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Fricas [B]  time = 1.63797, size = 838, normalized size = 3.31 \begin{align*} \frac{12 \, b^{5} e^{5} x^{5} + 48 \, a b^{4} e^{5} x^{4} - 3 \, b^{5} d^{5} - 5 \, a b^{4} d^{4} e - 10 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 125 \, a^{4} b d e^{4} - 77 \, a^{5} e^{5} - 24 \,{\left (5 \, b^{5} d^{2} e^{3} - 10 \, a b^{4} d e^{4} + 2 \, a^{2} b^{3} e^{5}\right )} x^{3} - 12 \,{\left (5 \, b^{5} d^{3} e^{2} + 15 \, a b^{4} d^{2} e^{3} - 45 \, a^{2} b^{3} d e^{4} + 21 \, a^{3} b^{2} e^{5}\right )} x^{2} - 4 \,{\left (5 \, b^{5} d^{4} e + 10 \, a b^{4} d^{3} e^{2} + 30 \, a^{2} b^{3} d^{2} e^{3} - 110 \, a^{3} b^{2} d e^{4} + 62 \, a^{4} b e^{5}\right )} x + 60 \,{\left (a^{4} b d e^{4} - a^{5} e^{5} +{\left (b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 4 \,{\left (a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 6 \,{\left (a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 4 \,{\left (a^{3} b^{2} d e^{4} - a^{4} b e^{5}\right )} x\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((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(12*b^5*e^5*x^5 + 48*a*b^4*e^5*x^4 - 3*b^5*d^5 - 5*a*b^4*d^4*e - 10*a^2*b^3*d^3*e^2 - 30*a^3*b^2*d^2*e^3
+ 125*a^4*b*d*e^4 - 77*a^5*e^5 - 24*(5*b^5*d^2*e^3 - 10*a*b^4*d*e^4 + 2*a^2*b^3*e^5)*x^3 - 12*(5*b^5*d^3*e^2 +
15*a*b^4*d^2*e^3 - 45*a^2*b^3*d*e^4 + 21*a^3*b^2*e^5)*x^2 - 4*(5*b^5*d^4*e + 10*a*b^4*d^3*e^2 + 30*a^2*b^3*d^
2*e^3 - 110*a^3*b^2*d*e^4 + 62*a^4*b*e^5)*x + 60*(a^4*b*d*e^4 - a^5*e^5 + (b^5*d*e^4 - a*b^4*e^5)*x^4 + 4*(a*b
^4*d*e^4 - a^2*b^3*e^5)*x^3 + 6*(a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 4*(a^3*b^2*d*e^4 - a^4*b*e^5)*x)*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{\left (d + e x\right )^{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((e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

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

[Out]

sage0*x