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

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

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Rubi [A]  time = 0.18, antiderivative size = 253, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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 {(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*}

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Mathematica [A]  time = 0.12, size = 242, normalized size = 0.96 \begin {gather*} \frac {-77 a^5 e^5+a^4 b e^4 (125 d-248 e x)-2 a^3 b^2 e^3 \left (15 d^2-220 d e x+126 e^2 x^2\right )-2 a^2 b^3 e^2 \left (5 d^3+60 d^2 e x-270 d e^2 x^2+24 e^3 x^3\right )+a b^4 e \left (-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\right )-60 e^4 (a+b x)^4 (a e-b d) \log (a+b x)-\left (b^5 \left (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\right )\right )}{12 b^6 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}

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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IntegrateAlgebraic [B]  time = 9.44, size = 5910, normalized size = 23.36 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [B]  time = 0.40, size = 412, normalized size = 1.63 \begin {gather*} \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 {gather*}

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

[Out]

sage0*x

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maple [B]  time = 0.07, size = 449, normalized size = 1.77 \begin {gather*} -\frac {\left (60 a \,b^{4} e^{5} x^{4} \ln \left (b x +a \right )-60 b^{5} d \,e^{4} x^{4} \ln \left (b x +a \right )-12 b^{5} e^{5} x^{5}+240 a^{2} b^{3} e^{5} x^{3} \ln \left (b x +a \right )-240 a \,b^{4} d \,e^{4} x^{3} \ln \left (b x +a \right )-48 a \,b^{4} e^{5} x^{4}+360 a^{3} b^{2} e^{5} x^{2} \ln \left (b x +a \right )-360 a^{2} b^{3} d \,e^{4} x^{2} \ln \left (b x +a \right )+48 a^{2} b^{3} e^{5} x^{3}-240 a \,b^{4} d \,e^{4} x^{3}+120 b^{5} d^{2} e^{3} x^{3}+240 a^{4} b \,e^{5} x \ln \left (b x +a \right )-240 a^{3} b^{2} d \,e^{4} x \ln \left (b x +a \right )+252 a^{3} b^{2} e^{5} x^{2}-540 a^{2} b^{3} d \,e^{4} x^{2}+180 a \,b^{4} d^{2} e^{3} x^{2}+60 b^{5} d^{3} e^{2} x^{2}+60 a^{5} e^{5} \ln \left (b x +a \right )-60 a^{4} b d \,e^{4} \ln \left (b x +a \right )+248 a^{4} b \,e^{5} x -440 a^{3} b^{2} d \,e^{4} x +120 a^{2} b^{3} d^{2} e^{3} x +40 a \,b^{4} d^{3} e^{2} x +20 b^{5} d^{4} e x +77 a^{5} e^{5}-125 a^{4} b d \,e^{4}+30 a^{3} b^{2} d^{2} e^{3}+10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e +3 b^{5} d^{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((e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

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

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maxima [B]  time = 1.54, size = 443, normalized size = 1.75 \begin {gather*} \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 {6 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{3}} - \frac {3 \, a^{3}}{b^{8} {\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}{b^{6} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {5}{6} \, d^{3} e^{2} {\left (\frac {6}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {8 \, a}{b^{6} {\left (x + \frac {a}{b}\right )}^{3}} + \frac {3 \, a^{2}}{b^{7} {\left (x + \frac {a}{b}\right )}^{4}}\right )} - \frac {d^{5}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \end {gather*}

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) + 6*a/(b^6*(x + a/b)^2) - 8*a^2/(b^7*(x + a/b)^3) - 3*a^3/(b^8*(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^6*(x + a/b)^4)) - 5/6*d^3*e^2*(6/(b^5*(x + a/b)^2) - 8*a/(b^6*(x + a/b)^
3) + 3*a^2/(b^7*(x + a/b)^4)) - 1/4*d^5/(b^5*(x + a/b)^4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^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((d + e*x)^5/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int((d + e*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 {\left (d + e x\right )^{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((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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