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

Optimal. Leaf size=373 \[ \frac {5 e^3 (a+b x) (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 e^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{3 b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^5}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 x (a+b x) (-5 a B e+A b e+5 b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^5 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.43, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \frac {e^4 x (a+b x) (-5 a B e+A b e+5 b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {10 e^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^3 (a+b x) (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{3 b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^5}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^5 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*e^2*(b*d - a*e)^2*(b*B*d + A*b*e - 2*a*B*e))/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(b*d - a*
e)^5)/(4*b^7*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((b*d - a*e)^4*(b*B*d + 5*A*b*e - 6*a*B*e))/(3*b^7*(
a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (5*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*a*B*e))/(2*b^7*(a + b*x)*S
qrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^4*(5*b*B*d + A*b*e - 5*a*B*e)*x*(a + b*x))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^
2]) + (B*e^5*x^2*(a + b*x))/(2*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*e^3*(b*d - a*e)*(2*b*B*d + A*b*e - 3*a*
B*e)*(a + b*x)*Log[a + b*x])/(b^7*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) (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 {(A+B x) (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^4 (5 b B d+A b e-5 a B e)}{b^{11}}+\frac {B e^5 x}{b^{10}}+\frac {(A b-a B) (b d-a e)^5}{b^{11} (a+b x)^5}+\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^{11} (a+b x)^4}+\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e)}{b^{11} (a+b x)^3}+\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e)}{b^{11} (a+b x)^2}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e)}{b^{11} (a+b x)}\right ) \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^5}{4 b^7 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^4 (b B d+5 A b e-6 a B e)}{3 b^7 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (b d-a e)^3 (b B d+2 A b e-3 a B e)}{2 b^7 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^4 (5 b B d+A b e-5 a B e) x (a+b x)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^5 x^2 (a+b x)}{2 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x) \log (a+b x)}{b^7 \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 513, normalized size = 1.38 \begin {gather*} \frac {-A b \left (77 a^5 e^5+a^4 b e^4 (248 e x-125 d)+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 )+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 )+B \left (171 a^6 e^5+7 a^5 b e^4 (72 e x-55 d)+2 a^4 b^2 e^3 \left (125 d^2-620 d e x+198 e^2 x^2\right )-2 a^3 b^3 e^2 \left (15 d^3-440 d^2 e x+630 d e^2 x^2+48 e^3 x^3\right )-a^2 b^4 e \left (5 d^4+120 d^3 e x-1080 d^2 e^2 x^2+240 d e^3 x^3+204 e^4 x^4\right )-a b^5 \left (d^5+20 d^4 e x+180 d^3 e^2 x^2-480 d^2 e^3 x^3-240 d e^4 x^4+36 e^5 x^5\right )+2 b^6 x \left (-2 d^5-15 d^4 e x-60 d^3 e^2 x^2+30 d e^4 x^4+3 e^5 x^5\right )\right )+60 e^3 (a+b x)^4 (b d-a e) \log (a+b x) (-3 a B e+A b e+2 b B d)}{12 b^7 (a+b x)^3 \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(A*b*(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))) + B*(171
*a^6*e^5 + 7*a^5*b*e^4*(-55*d + 72*e*x) + 2*a^4*b^2*e^3*(125*d^2 - 620*d*e*x + 198*e^2*x^2) - 2*a^3*b^3*e^2*(1
5*d^3 - 440*d^2*e*x + 630*d*e^2*x^2 + 48*e^3*x^3) - a^2*b^4*e*(5*d^4 + 120*d^3*e*x - 1080*d^2*e^2*x^2 + 240*d*
e^3*x^3 + 204*e^4*x^4) + 2*b^6*x*(-2*d^5 - 15*d^4*e*x - 60*d^3*e^2*x^2 + 30*d*e^4*x^4 + 3*e^5*x^5) - a*b^5*(d^
5 + 20*d^4*e*x + 180*d^3*e^2*x^2 - 480*d^2*e^3*x^3 - 240*d*e^4*x^4 + 36*e^5*x^5)) + 60*e^3*(b*d - a*e)*(2*b*B*
d + A*b*e - 3*a*B*e)*(a + b*x)^4*Log[a + b*x])/(12*b^7*(a + b*x)^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [B]  time = 48.39, size = 14568, normalized size = 39.06 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(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 = 0.45, size = 914, normalized size = 2.45 \begin {gather*} \frac {6 \, B b^{6} e^{5} x^{6} - {\left (B a b^{5} + 3 \, A b^{6}\right )} d^{5} - 5 \, {\left (B a^{2} b^{4} + A a b^{5}\right )} d^{4} e - 10 \, {\left (3 \, B a^{3} b^{3} + A a^{2} b^{4}\right )} d^{3} e^{2} + 10 \, {\left (25 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} d^{2} e^{3} - 5 \, {\left (77 \, B a^{5} b - 25 \, A a^{4} b^{2}\right )} d e^{4} + {\left (171 \, B a^{6} - 77 \, A a^{5} b\right )} e^{5} + 12 \, {\left (5 \, B b^{6} d e^{4} - {\left (3 \, B a b^{5} - A b^{6}\right )} e^{5}\right )} x^{5} + 12 \, {\left (20 \, B a b^{5} d e^{4} - {\left (17 \, B a^{2} b^{4} - 4 \, A a b^{5}\right )} e^{5}\right )} x^{4} - 24 \, {\left (5 \, B b^{6} d^{3} e^{2} - 5 \, {\left (4 \, B a b^{5} - A b^{6}\right )} d^{2} e^{3} + 10 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} d e^{4} + 2 \, {\left (2 \, B a^{3} b^{3} + A a^{2} b^{4}\right )} e^{5}\right )} x^{3} - 6 \, {\left (5 \, B b^{6} d^{4} e + 10 \, {\left (3 \, B a b^{5} + A b^{6}\right )} d^{3} e^{2} - 30 \, {\left (6 \, B a^{2} b^{4} - A a b^{5}\right )} d^{2} e^{3} + 30 \, {\left (7 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d e^{4} - 6 \, {\left (11 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} e^{5}\right )} x^{2} - 4 \, {\left (B b^{6} d^{5} + 5 \, {\left (B a b^{5} + A b^{6}\right )} d^{4} e + 10 \, {\left (3 \, B a^{2} b^{4} + A a b^{5}\right )} d^{3} e^{2} - 10 \, {\left (22 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 10 \, {\left (31 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} d e^{4} - 2 \, {\left (63 \, B a^{5} b - 31 \, A a^{4} b^{2}\right )} e^{5}\right )} x + 60 \, {\left (2 \, B a^{4} b^{2} d^{2} e^{3} - {\left (5 \, B a^{5} b - A a^{4} b^{2}\right )} d e^{4} + {\left (3 \, B a^{6} - A a^{5} b\right )} e^{5} + {\left (2 \, B b^{6} d^{2} e^{3} - {\left (5 \, B a b^{5} - A b^{6}\right )} d e^{4} + {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} e^{5}\right )} x^{4} + 4 \, {\left (2 \, B a b^{5} d^{2} e^{3} - {\left (5 \, B a^{2} b^{4} - A a b^{5}\right )} d e^{4} + {\left (3 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 6 \, {\left (2 \, B a^{2} b^{4} d^{2} e^{3} - {\left (5 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} d e^{4} + {\left (3 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} e^{5}\right )} x^{2} + 4 \, {\left (2 \, B a^{3} b^{3} d^{2} e^{3} - {\left (5 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} d e^{4} + {\left (3 \, B a^{5} b - A a^{4} b^{2}\right )} e^{5}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*B*b^6*e^5*x^6 - (B*a*b^5 + 3*A*b^6)*d^5 - 5*(B*a^2*b^4 + A*a*b^5)*d^4*e - 10*(3*B*a^3*b^3 + A*a^2*b^4)
*d^3*e^2 + 10*(25*B*a^4*b^2 - 3*A*a^3*b^3)*d^2*e^3 - 5*(77*B*a^5*b - 25*A*a^4*b^2)*d*e^4 + (171*B*a^6 - 77*A*a
^5*b)*e^5 + 12*(5*B*b^6*d*e^4 - (3*B*a*b^5 - A*b^6)*e^5)*x^5 + 12*(20*B*a*b^5*d*e^4 - (17*B*a^2*b^4 - 4*A*a*b^
5)*e^5)*x^4 - 24*(5*B*b^6*d^3*e^2 - 5*(4*B*a*b^5 - A*b^6)*d^2*e^3 + 10*(B*a^2*b^4 - A*a*b^5)*d*e^4 + 2*(2*B*a^
3*b^3 + A*a^2*b^4)*e^5)*x^3 - 6*(5*B*b^6*d^4*e + 10*(3*B*a*b^5 + A*b^6)*d^3*e^2 - 30*(6*B*a^2*b^4 - A*a*b^5)*d
^2*e^3 + 30*(7*B*a^3*b^3 - 3*A*a^2*b^4)*d*e^4 - 6*(11*B*a^4*b^2 - 7*A*a^3*b^3)*e^5)*x^2 - 4*(B*b^6*d^5 + 5*(B*
a*b^5 + A*b^6)*d^4*e + 10*(3*B*a^2*b^4 + A*a*b^5)*d^3*e^2 - 10*(22*B*a^3*b^3 - 3*A*a^2*b^4)*d^2*e^3 + 10*(31*B
*a^4*b^2 - 11*A*a^3*b^3)*d*e^4 - 2*(63*B*a^5*b - 31*A*a^4*b^2)*e^5)*x + 60*(2*B*a^4*b^2*d^2*e^3 - (5*B*a^5*b -
A*a^4*b^2)*d*e^4 + (3*B*a^6 - A*a^5*b)*e^5 + (2*B*b^6*d^2*e^3 - (5*B*a*b^5 - A*b^6)*d*e^4 + (3*B*a^2*b^4 - A*
a*b^5)*e^5)*x^4 + 4*(2*B*a*b^5*d^2*e^3 - (5*B*a^2*b^4 - A*a*b^5)*d*e^4 + (3*B*a^3*b^3 - A*a^2*b^4)*e^5)*x^3 +
6*(2*B*a^2*b^4*d^2*e^3 - (5*B*a^3*b^3 - A*a^2*b^4)*d*e^4 + (3*B*a^4*b^2 - A*a^3*b^3)*e^5)*x^2 + 4*(2*B*a^3*b^3
*d^2*e^3 - (5*B*a^4*b^2 - A*a^3*b^3)*d*e^4 + (3*B*a^5*b - A*a^4*b^2)*e^5)*x)*log(b*x + a))/(b^11*x^4 + 4*a*b^1
0*x^3 + 6*a^2*b^9*x^2 + 4*a^3*b^8*x + a^4*b^7)

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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((B*x+A)*(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 = 1153, normalized size = 3.09 \begin {gather*} -\frac {\left (-6 B \,b^{6} e^{5} x^{6}+60 A a \,b^{5} e^{5} x^{4} \ln \left (b x +a \right )-60 A \,b^{6} d \,e^{4} x^{4} \ln \left (b x +a \right )-12 A \,b^{6} e^{5} x^{5}-180 B \,a^{2} b^{4} e^{5} x^{4} \ln \left (b x +a \right )+300 B a \,b^{5} d \,e^{4} x^{4} \ln \left (b x +a \right )+36 B a \,b^{5} e^{5} x^{5}-120 B \,b^{6} d^{2} e^{3} x^{4} \ln \left (b x +a \right )-60 B \,b^{6} d \,e^{4} x^{5}+240 A \,a^{2} b^{4} e^{5} x^{3} \ln \left (b x +a \right )-240 A a \,b^{5} d \,e^{4} x^{3} \ln \left (b x +a \right )-48 A a \,b^{5} e^{5} x^{4}-720 B \,a^{3} b^{3} e^{5} x^{3} \ln \left (b x +a \right )+1200 B \,a^{2} b^{4} d \,e^{4} x^{3} \ln \left (b x +a \right )+204 B \,a^{2} b^{4} e^{5} x^{4}-480 B a \,b^{5} d^{2} e^{3} x^{3} \ln \left (b x +a \right )-240 B a \,b^{5} d \,e^{4} x^{4}+360 A \,a^{3} b^{3} e^{5} x^{2} \ln \left (b x +a \right )-360 A \,a^{2} b^{4} d \,e^{4} x^{2} \ln \left (b x +a \right )+48 A \,a^{2} b^{4} e^{5} x^{3}-240 A a \,b^{5} d \,e^{4} x^{3}+120 A \,b^{6} d^{2} e^{3} x^{3}-1080 B \,a^{4} b^{2} e^{5} x^{2} \ln \left (b x +a \right )+1800 B \,a^{3} b^{3} d \,e^{4} x^{2} \ln \left (b x +a \right )+96 B \,a^{3} b^{3} e^{5} x^{3}-720 B \,a^{2} b^{4} d^{2} e^{3} x^{2} \ln \left (b x +a \right )+240 B \,a^{2} b^{4} d \,e^{4} x^{3}-480 B a \,b^{5} d^{2} e^{3} x^{3}+120 B \,b^{6} d^{3} e^{2} x^{3}+240 A \,a^{4} b^{2} e^{5} x \ln \left (b x +a \right )-240 A \,a^{3} b^{3} d \,e^{4} x \ln \left (b x +a \right )+252 A \,a^{3} b^{3} e^{5} x^{2}-540 A \,a^{2} b^{4} d \,e^{4} x^{2}+180 A a \,b^{5} d^{2} e^{3} x^{2}+60 A \,b^{6} d^{3} e^{2} x^{2}-720 B \,a^{5} b \,e^{5} x \ln \left (b x +a \right )+1200 B \,a^{4} b^{2} d \,e^{4} x \ln \left (b x +a \right )-396 B \,a^{4} b^{2} e^{5} x^{2}-480 B \,a^{3} b^{3} d^{2} e^{3} x \ln \left (b x +a \right )+1260 B \,a^{3} b^{3} d \,e^{4} x^{2}-1080 B \,a^{2} b^{4} d^{2} e^{3} x^{2}+180 B a \,b^{5} d^{3} e^{2} x^{2}+30 B \,b^{6} d^{4} e \,x^{2}+60 A \,a^{5} b \,e^{5} \ln \left (b x +a \right )-60 A \,a^{4} b^{2} d \,e^{4} \ln \left (b x +a \right )+248 A \,a^{4} b^{2} e^{5} x -440 A \,a^{3} b^{3} d \,e^{4} x +120 A \,a^{2} b^{4} d^{2} e^{3} x +40 A a \,b^{5} d^{3} e^{2} x +20 A \,b^{6} d^{4} e x -180 B \,a^{6} e^{5} \ln \left (b x +a \right )+300 B \,a^{5} b d \,e^{4} \ln \left (b x +a \right )-504 B \,a^{5} b \,e^{5} x -120 B \,a^{4} b^{2} d^{2} e^{3} \ln \left (b x +a \right )+1240 B \,a^{4} b^{2} d \,e^{4} x -880 B \,a^{3} b^{3} d^{2} e^{3} x +120 B \,a^{2} b^{4} d^{3} e^{2} x +20 B a \,b^{5} d^{4} e x +4 B \,b^{6} d^{5} x +77 A \,a^{5} b \,e^{5}-125 A \,a^{4} b^{2} d \,e^{4}+30 A \,a^{3} b^{3} d^{2} e^{3}+10 A \,a^{2} b^{4} d^{3} e^{2}+5 A a \,b^{5} d^{4} e +3 A \,b^{6} d^{5}-171 B \,a^{6} e^{5}+385 B \,a^{5} b d \,e^{4}-250 B \,a^{4} b^{2} d^{2} e^{3}+30 B \,a^{3} b^{3} d^{3} e^{2}+5 B \,a^{2} b^{4} d^{4} e +B a \,b^{5} d^{5}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/12*(77*A*a^5*b*e^5-250*B*a^4*b^2*d^2*e^3+120*B*x*a^2*b^4*d^3*e^2+385*B*a^5*b*d*e^4+3*A*b^6*d^5-171*B*a^6*e^
5-12*A*x^5*b^6*e^5-180*B*ln(b*x+a)*a^6*e^5-125*A*a^4*b^2*d*e^4+30*B*a^3*b^3*d^3*e^2+5*a^2*B*b^4*d^4*e+B*a*b^5*
d^5+30*A*a^3*b^3*d^2*e^3-480*B*ln(b*x+a)*x^3*a*b^5*d^2*e^3-240*A*ln(b*x+a)*x^3*a*b^5*d*e^4-240*A*ln(b*x+a)*x*a
^3*b^3*d*e^4+300*B*ln(b*x+a)*x^4*a*b^5*d*e^4+1200*B*ln(b*x+a)*x*a^4*b^2*d*e^4-480*B*ln(b*x+a)*x*a^3*b^3*d^2*e^
3+4*B*x*b^6*d^5-6*B*x^6*b^6*e^5+10*A*a^2*b^4*d^3*e^2-360*A*ln(b*x+a)*x^2*a^2*b^4*d*e^4+1800*B*ln(b*x+a)*x^2*a^
3*b^3*d*e^4-720*B*ln(b*x+a)*x^2*a^2*b^4*d^2*e^3+1200*B*ln(b*x+a)*x^3*a^2*b^4*d*e^4+5*A*a*b^5*d^4*e-120*B*ln(b*
x+a)*x^4*b^6*d^2*e^3+240*A*ln(b*x+a)*x^3*a^2*b^4*e^5-60*A*ln(b*x+a)*a^4*b^2*d*e^4+300*B*ln(b*x+a)*a^5*b*d*e^4-
120*B*ln(b*x+a)*a^4*b^2*d^2*e^3+60*A*ln(b*x+a)*x^4*a*b^5*e^5-60*A*ln(b*x+a)*x^4*b^6*d*e^4-180*B*ln(b*x+a)*x^4*
a^2*b^4*e^5+1240*B*x*a^4*b^2*d*e^4-240*B*x^4*a*b^5*d*e^4+240*B*x^3*a^2*b^4*d*e^4+1260*B*x^2*a^3*b^3*d*e^4+120*
A*x*a^2*b^4*d^2*e^3+40*A*x*a*b^5*d^3*e^2-880*B*x*a^3*b^3*d^2*e^3-48*A*x^4*a*b^5*e^5+60*A*ln(b*x+a)*a^5*b*e^5+3
6*B*x^5*a*b^5*e^5-60*B*x^5*b^6*d*e^4-504*B*x*a^5*b*e^5+20*A*x*b^6*d^4*e+60*A*x^2*b^6*d^3*e^2-396*B*x^2*a^4*b^2
*e^5+30*B*x^2*b^6*d^4*e+248*A*x*a^4*b^2*e^5+96*B*x^3*a^3*b^3*e^5+120*B*x^3*b^6*d^3*e^2+252*A*x^2*a^3*b^3*e^5+1
20*A*x^3*b^6*d^2*e^3+204*B*x^4*a^2*b^4*e^5+48*A*x^3*a^2*b^4*e^5-540*A*x^2*a^2*b^4*d*e^4+180*A*x^2*a*b^5*d^2*e^
3-240*A*x^3*a*b^5*d*e^4-1080*B*x^2*a^2*b^4*d^2*e^3+180*B*x^2*a*b^5*d^3*e^2-480*B*x^3*a*b^5*d^2*e^3-440*A*x*a^3
*b^3*d*e^4-720*B*ln(b*x+a)*x^3*a^3*b^3*e^5+360*A*ln(b*x+a)*x^2*a^3*b^3*e^5-1080*B*ln(b*x+a)*x^2*a^4*b^2*e^5+24
0*A*ln(b*x+a)*x*a^4*b^2*e^5-720*B*ln(b*x+a)*x*a^5*b*e^5+20*B*x*a*b^5*d^4*e)*(b*x+a)/b^7/((b*x+a)^2)^(5/2)

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maxima [B]  time = 1.19, size = 1010, normalized size = 2.71

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*e^5*((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^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) + 60*a^2*log(b*x + a)/b^7) + 5/12*B*d*e^4
*((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) + 1/12*A*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) + 5/6*B*d^2*e^3*((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) + 5/12*A*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*B*d^3*e^2*(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)) - 5/6*A*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)) - 1/12*B*d^5*(4/((b^2*x^2 + 2*a*b*x + a^2)^(3
/2)*b^2) - 3*a/(b^6*(x + a/b)^4)) - 5/12*A*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)) - 5/12*B*d^4*e*(6/(b^5*(x + a/b)^2) - 8*a/(b^6*(x + a/b)^3) + 3*a^2/(b^7*(x + a/b)^4)) - 5/6*A*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*A*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 (A+B\,x\right )\,{\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(((A + B*x)*(d + e*x)^5)/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int(((A + B*x)*(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 (A + B x\right ) \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((B*x+A)*(e*x+d)**5/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**5/((a + b*x)**2)**(5/2), x)

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