3.1776 \(\int \frac{(A+B x) (d+e x)^4}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=310 \[ -\frac{2 e^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 (a+b x) \log (a+b x) (-5 a B e+A b e+4 b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (b d-a e)^4}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^4 x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(-2*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(b*d - a
*e)^4)/(4*b^6*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e))/(3*b^6*
(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e))/(b^6*(a + b*x)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) + (B*e^4*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^3*(4*b*B*d + A*b*e
- 5*a*B*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 = 0.303944, antiderivative size = 310, normalized size of antiderivative = 1., 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} \[ -\frac{2 e^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 (a+b x) \log (a+b x) (-5 a B e+A b e+4 b B d)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (b d-a e)^4}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^4 x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-2*e^2*(b*d - a*e)*(3*b*B*d + 2*A*b*e - 5*a*B*e))/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*B)*(b*d - a
*e)^4)/(4*b^6*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e))/(3*b^6*
(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e))/(b^6*(a + b*x)*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) + (B*e^4*x*(a + b*x))/(b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^3*(4*b*B*d + A*b*e
- 5*a*B*e)*(a + b*x)*Log[a + b*x])/(b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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]

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]))))

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^4}{\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)^4}{\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{B e^4}{b^{10}}+\frac{(A b-a B) (b d-a e)^4}{b^{10} (a+b x)^5}+\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^{10} (a+b x)^4}+\frac{2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^{10} (a+b x)^3}+\frac{2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e)}{b^{10} (a+b x)^2}+\frac{e^3 (4 b B d+A b e-5 a B e)}{b^{10} (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{2 e^2 (b d-a e) (3 b B d+2 A b e-5 a B e)}{b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (b d-a e)^4}{4 b^6 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(b d-a e)^3 (b B d+4 A b e-5 a B e)}{3 b^6 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{B e^4 x (a+b x)}{b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{e^3 (4 b B d+A b e-5 a B 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 = 0.235358, size = 331, normalized size = 1.07 \[ \frac{-A b (b d-a e) \left (a^2 b e^2 (13 d+88 e x)+25 a^3 e^3+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (16 d^2 e x+3 d^3+36 d e^2 x^2+48 e^3 x^3\right )\right )-B \left (2 a^3 b^2 e^2 \left (9 d^2-176 d e x+126 e^2 x^2\right )+4 a^2 b^3 e \left (18 d^2 e x+d^3-108 d e^2 x^2+12 e^3 x^3\right )+4 a^4 b e^3 (62 e x-25 d)+77 a^5 e^4+a b^4 \left (108 d^2 e^2 x^2+16 d^3 e x+d^4-192 d e^3 x^3-48 e^4 x^4\right )+4 b^5 x \left (18 d^2 e^2 x^2+6 d^3 e x+d^4-3 e^4 x^4\right )\right )+12 e^3 (a+b x)^4 \log (a+b x) (-5 a B e+A b e+4 b B d)}{12 b^6 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(-(A*b*(b*d - a*e)*(25*a^3*e^3 + a^2*b*e^2*(13*d + 88*e*x) + a*b^2*e*(7*d^2 + 40*d*e*x + 108*e^2*x^2) + b^3*(3
*d^3 + 16*d^2*e*x + 36*d*e^2*x^2 + 48*e^3*x^3))) - B*(77*a^5*e^4 + 4*a^4*b*e^3*(-25*d + 62*e*x) + 2*a^3*b^2*e^
2*(9*d^2 - 176*d*e*x + 126*e^2*x^2) + 4*a^2*b^3*e*(d^3 + 18*d^2*e*x - 108*d*e^2*x^2 + 12*e^3*x^3) + a*b^4*(d^4
+ 16*d^3*e*x + 108*d^2*e^2*x^2 - 192*d*e^3*x^3 - 48*e^4*x^4) + 4*b^5*x*(d^4 + 6*d^3*e*x + 18*d^2*e^2*x^2 - 3*
e^4*x^4)) + 12*e^3*(4*b*B*d + A*b*e - 5*a*B*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 = 0.017, size = 735, normalized size = 2.4 \begin{align*}{\frac{ \left ( -72\,A{x}^{2}a{b}^{4}d{e}^{3}-3\,A{b}^{5}{d}^{4}-77\,B{a}^{5}{e}^{4}+192\,B\ln \left ( bx+a \right ){x}^{3}a{b}^{4}d{e}^{3}-12\,A{a}^{3}{b}^{2}d{e}^{3}+100\,B{a}^{4}bd{e}^{3}-18\,B{a}^{3}{b}^{2}{d}^{2}{e}^{2}+72\,A\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}{e}^{4}+192\,B{x}^{3}a{b}^{4}d{e}^{3}-24\,Axa{b}^{4}{d}^{2}{e}^{2}-16\,Bxa{b}^{4}{d}^{3}e+432\,B{x}^{2}{a}^{2}{b}^{3}d{e}^{3}-108\,B{x}^{2}a{b}^{4}{d}^{2}{e}^{2}-48\,Ax{a}^{2}{b}^{3}d{e}^{3}+352\,Bx{a}^{3}{b}^{2}d{e}^{3}-72\,Bx{a}^{2}{b}^{3}{d}^{2}{e}^{2}+48\,B\ln \left ( bx+a \right ){a}^{4}bd{e}^{3}-360\,B\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}{e}^{4}+48\,A\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}{e}^{4}-240\,B\ln \left ( bx+a \right ) x{a}^{4}b{e}^{4}+48\,A\ln \left ( bx+a \right ){x}^{3}a{b}^{4}{e}^{4}-240\,B\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}{e}^{4}-60\,B\ln \left ( bx+a \right ){x}^{4}a{b}^{4}{e}^{4}+48\,B\ln \left ( bx+a \right ){x}^{4}{b}^{5}d{e}^{3}+12\,B{x}^{5}{b}^{5}{e}^{4}-60\,B\ln \left ( bx+a \right ){a}^{5}{e}^{4}-4\,Bx{b}^{5}{d}^{4}-Ba{b}^{4}{d}^{4}+25\,A{a}^{4}b{e}^{4}+108\,A{x}^{2}{a}^{2}{b}^{3}{e}^{4}+12\,A\ln \left ( bx+a \right ){a}^{4}b{e}^{4}+48\,A{x}^{3}a{b}^{4}{e}^{4}-48\,A{x}^{3}{b}^{5}d{e}^{3}-48\,B{x}^{3}{a}^{2}{b}^{3}{e}^{4}-72\,B{x}^{3}{b}^{5}{d}^{2}{e}^{2}+48\,B{x}^{4}a{b}^{4}{e}^{4}-252\,B{x}^{2}{a}^{3}{b}^{2}{e}^{4}+88\,Ax{a}^{3}{b}^{2}{e}^{4}-16\,Ax{b}^{5}{d}^{3}e-248\,Bx{a}^{4}b{e}^{4}+12\,A\ln \left ( bx+a \right ){x}^{4}{b}^{5}{e}^{4}-36\,A{x}^{2}{b}^{5}{d}^{2}{e}^{2}-24\,B{x}^{2}{b}^{5}{d}^{3}e-4\,Aa{b}^{4}{d}^{3}e-4\,{b}^{3}B{a}^{2}{d}^{3}e-6\,A{a}^{2}{b}^{3}{d}^{2}{e}^{2}+192\,B\ln \left ( bx+a \right ) x{a}^{3}{b}^{2}d{e}^{3}+288\,B\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{3}d{e}^{3} \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((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/12*(-72*A*x^2*a*b^4*d*e^3-3*A*b^5*d^4-77*B*a^5*e^4+192*B*ln(b*x+a)*x^3*a*b^4*d*e^3-12*A*a^3*b^2*d*e^3+100*B*
a^4*b*d*e^3-18*B*a^3*b^2*d^2*e^2+72*A*ln(b*x+a)*x^2*a^2*b^3*e^4+192*B*x^3*a*b^4*d*e^3-24*A*x*a*b^4*d^2*e^2-16*
B*x*a*b^4*d^3*e+432*B*x^2*a^2*b^3*d*e^3-108*B*x^2*a*b^4*d^2*e^2-48*A*x*a^2*b^3*d*e^3+352*B*x*a^3*b^2*d*e^3-72*
B*x*a^2*b^3*d^2*e^2+48*B*ln(b*x+a)*a^4*b*d*e^3-360*B*ln(b*x+a)*x^2*a^3*b^2*e^4+48*A*ln(b*x+a)*x*a^3*b^2*e^4-24
0*B*ln(b*x+a)*x*a^4*b*e^4+48*A*ln(b*x+a)*x^3*a*b^4*e^4-240*B*ln(b*x+a)*x^3*a^2*b^3*e^4-60*B*ln(b*x+a)*x^4*a*b^
4*e^4+48*B*ln(b*x+a)*x^4*b^5*d*e^3+12*B*x^5*b^5*e^4-60*B*ln(b*x+a)*a^5*e^4-4*B*x*b^5*d^4-B*a*b^4*d^4+25*A*a^4*
b*e^4+108*A*x^2*a^2*b^3*e^4+12*A*ln(b*x+a)*a^4*b*e^4+48*A*x^3*a*b^4*e^4-48*A*x^3*b^5*d*e^3-48*B*x^3*a^2*b^3*e^
4-72*B*x^3*b^5*d^2*e^2+48*B*x^4*a*b^4*e^4-252*B*x^2*a^3*b^2*e^4+88*A*x*a^3*b^2*e^4-16*A*x*b^5*d^3*e-248*B*x*a^
4*b*e^4+12*A*ln(b*x+a)*x^4*b^5*e^4-36*A*x^2*b^5*d^2*e^2-24*B*x^2*b^5*d^3*e-4*A*a*b^4*d^3*e-4*b^3*B*a^2*d^3*e-6
*A*a^2*b^3*d^2*e^2+192*B*ln(b*x+a)*x*a^3*b^2*d*e^3+288*B*ln(b*x+a)*x^2*a^2*b^3*d*e^3)*(b*x+a)/b^6/((b*x+a)^2)^
(5/2)

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Maxima [B]  time = 1.19119, size = 1149, normalized size = 3.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

1/12*B*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) + 1/3*B*d*e^3*((48*a*b^3*x^3 + 1
08*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) + 1/12*A*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) - 1/2*B*d^2*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) + 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)) - 1/3*A*d*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)) - 1/12*B*d^4*(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)) - 1/3*A*d^
3*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)) - 1/3*B*d^3*e*(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)) - 1/2*A*d^2*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*A*d^4/((b^
2)^(5/2)*(x + a/b)^4)

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Fricas [B]  time = 1.51598, size = 1251, normalized size = 4.04 \begin{align*} \frac{12 \, B b^{5} e^{4} x^{5} + 48 \, B a b^{4} e^{4} x^{4} -{\left (B a b^{4} + 3 \, A b^{5}\right )} d^{4} - 4 \,{\left (B a^{2} b^{3} + A a b^{4}\right )} d^{3} e - 6 \,{\left (3 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{2} + 4 \,{\left (25 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} d e^{3} -{\left (77 \, B a^{5} - 25 \, A a^{4} b\right )} e^{4} - 24 \,{\left (3 \, B b^{5} d^{2} e^{2} - 2 \,{\left (4 \, B a b^{4} - A b^{5}\right )} d e^{3} + 2 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} - 12 \,{\left (2 \, B b^{5} d^{3} e + 3 \,{\left (3 \, B a b^{4} + A b^{5}\right )} d^{2} e^{2} - 6 \,{\left (6 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} + 3 \,{\left (7 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 4 \,{\left (B b^{5} d^{4} + 4 \,{\left (B a b^{4} + A b^{5}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{2} - 4 \,{\left (22 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} + 2 \,{\left (31 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \,{\left (4 \, B a^{4} b d e^{3} -{\left (5 \, B a^{5} - A a^{4} b\right )} e^{4} +{\left (4 \, B b^{5} d e^{3} -{\left (5 \, B a b^{4} - A b^{5}\right )} e^{4}\right )} x^{4} + 4 \,{\left (4 \, B a b^{4} d e^{3} -{\left (5 \, B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} + 6 \,{\left (4 \, B a^{2} b^{3} d e^{3} -{\left (5 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 4 \,{\left (4 \, B a^{3} b^{2} d e^{3} -{\left (5 \, B a^{4} b - A a^{3} b^{2}\right )} e^{4}\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((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(12*B*b^5*e^4*x^5 + 48*B*a*b^4*e^4*x^4 - (B*a*b^4 + 3*A*b^5)*d^4 - 4*(B*a^2*b^3 + A*a*b^4)*d^3*e - 6*(3*B
*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 4*(25*B*a^4*b - 3*A*a^3*b^2)*d*e^3 - (77*B*a^5 - 25*A*a^4*b)*e^4 - 24*(3*B*b^5
*d^2*e^2 - 2*(4*B*a*b^4 - A*b^5)*d*e^3 + 2*(B*a^2*b^3 - A*a*b^4)*e^4)*x^3 - 12*(2*B*b^5*d^3*e + 3*(3*B*a*b^4 +
A*b^5)*d^2*e^2 - 6*(6*B*a^2*b^3 - A*a*b^4)*d*e^3 + 3*(7*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 - 4*(B*b^5*d^4 + 4*
(B*a*b^4 + A*b^5)*d^3*e + 6*(3*B*a^2*b^3 + A*a*b^4)*d^2*e^2 - 4*(22*B*a^3*b^2 - 3*A*a^2*b^3)*d*e^3 + 2*(31*B*a
^4*b - 11*A*a^3*b^2)*e^4)*x + 12*(4*B*a^4*b*d*e^3 - (5*B*a^5 - A*a^4*b)*e^4 + (4*B*b^5*d*e^3 - (5*B*a*b^4 - A*
b^5)*e^4)*x^4 + 4*(4*B*a*b^4*d*e^3 - (5*B*a^2*b^3 - A*a*b^4)*e^4)*x^3 + 6*(4*B*a^2*b^3*d*e^3 - (5*B*a^3*b^2 -
A*a^2*b^3)*e^4)*x^2 + 4*(4*B*a^3*b^2*d*e^3 - (5*B*a^4*b - A*a^3*b^2)*e^4)*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 (A + B x\right ) \left (d + e x\right )^{4}}{\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((B*x+A)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**4/((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((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x