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

[Out]

-2*e^2*(-a*e+b*d)*(2*A*b*e-5*B*a*e+3*B*b*d)/b^6/((b*x+a)^2)^(1/2)-1/4*(A*b-B*a)*(-a*e+b*d)^4/b^6/(b*x+a)^3/((b
*x+a)^2)^(1/2)-1/3*(-a*e+b*d)^3*(4*A*b*e-5*B*a*e+B*b*d)/b^6/(b*x+a)^2/((b*x+a)^2)^(1/2)-e*(-a*e+b*d)^2*(3*A*b*
e-5*B*a*e+2*B*b*d)/b^6/(b*x+a)/((b*x+a)^2)^(1/2)+B*e^4*x*(b*x+a)/b^5/((b*x+a)^2)^(1/2)+e^3*(A*b*e-5*B*a*e+4*B*
b*d)*(b*x+a)*ln(b*x+a)/b^6/((b*x+a)^2)^(1/2)

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Rubi [A]  time = 0.30, antiderivative size = 310, 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} \[ -\frac {2 e^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 (a+b x) \log (a+b x) (-5 a B e+A b e+4 b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{3 b^6 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^4}{4 b^6 (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^4 x (a+b x)}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}} \]

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 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)^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*}

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Mathematica [A]  time = 0.22, size = 331, normalized size = 1.07 \[ \frac {-A b (b d-a e) \left (25 a^3 e^3+a^2 b e^2 (13 d+88 e x)+a b^2 e \left (7 d^2+40 d e x+108 e^2 x^2\right )+b^3 \left (3 d^3+16 d^2 e x+36 d e^2 x^2+48 e^3 x^3\right )\right )-B \left (77 a^5 e^4+4 a^4 b e^3 (62 e x-25 d)+2 a^3 b^2 e^2 \left (9 d^2-176 d e x+126 e^2 x^2\right )+4 a^2 b^3 e \left (d^3+18 d^2 e x-108 d e^2 x^2+12 e^3 x^3\right )+a b^4 \left (d^4+16 d^3 e x+108 d^2 e^2 x^2-192 d e^3 x^3-48 e^4 x^4\right )+4 b^5 x \left (d^4+6 d^3 e x+18 d^2 e^2 x^2-3 e^4 x^4\right )\right )+12 e^3 (a+b x)^4 \log (a+b x) (-5 a B e+A b e+4 b B d)}{12 b^6 (a+b x)^3 \sqrt {(a+b x)^2}} \]

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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fricas [B]  time = 0.54, size = 619, normalized size = 2.00 \[ \frac {12 \, B b^{5} e^{4} x^{5} + 48 \, B a b^{4} e^{4} x^{4} - {\left (B a b^{4} + 3 \, A b^{5}\right )} d^{4} - 4 \, {\left (B a^{2} b^{3} + A a b^{4}\right )} d^{3} e - 6 \, {\left (3 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{2} + 4 \, {\left (25 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} d e^{3} - {\left (77 \, B a^{5} - 25 \, A a^{4} b\right )} e^{4} - 24 \, {\left (3 \, B b^{5} d^{2} e^{2} - 2 \, {\left (4 \, B a b^{4} - A b^{5}\right )} d e^{3} + 2 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} - 12 \, {\left (2 \, B b^{5} d^{3} e + 3 \, {\left (3 \, B a b^{4} + A b^{5}\right )} d^{2} e^{2} - 6 \, {\left (6 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} + 3 \, {\left (7 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 4 \, {\left (B b^{5} d^{4} + 4 \, {\left (B a b^{4} + A b^{5}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (22 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} + 2 \, {\left (31 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (4 \, B a^{4} b d e^{3} - {\left (5 \, B a^{5} - A a^{4} b\right )} e^{4} + {\left (4 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - A b^{5}\right )} e^{4}\right )} x^{4} + 4 \, {\left (4 \, B a b^{4} d e^{3} - {\left (5 \, B a^{2} b^{3} - A a b^{4}\right )} e^{4}\right )} x^{3} + 6 \, {\left (4 \, B a^{2} b^{3} d e^{3} - {\left (5 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 4 \, {\left (4 \, B a^{3} b^{2} d e^{3} - {\left (5 \, B a^{4} b - A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{10} x^{4} + 4 \, a b^{9} x^{3} + 6 \, a^{2} b^{8} x^{2} + 4 \, a^{3} b^{7} x + a^{4} b^{6}\right )}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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

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maple [B]  time = 0.08, size = 735, normalized size = 2.37 \[ \frac {\left (12 A \,b^{5} e^{4} x^{4} \ln \left (b x +a \right )-60 B a \,b^{4} e^{4} x^{4} \ln \left (b x +a \right )+48 B \,b^{5} d \,e^{3} x^{4} \ln \left (b x +a \right )+12 B \,b^{5} e^{4} x^{5}+48 A a \,b^{4} e^{4} x^{3} \ln \left (b x +a \right )-240 B \,a^{2} b^{3} e^{4} x^{3} \ln \left (b x +a \right )+192 B a \,b^{4} d \,e^{3} x^{3} \ln \left (b x +a \right )+48 B a \,b^{4} e^{4} x^{4}+72 A \,a^{2} b^{3} e^{4} x^{2} \ln \left (b x +a \right )+48 A a \,b^{4} e^{4} x^{3}-48 A \,b^{5} d \,e^{3} x^{3}-360 B \,a^{3} b^{2} e^{4} x^{2} \ln \left (b x +a \right )+288 B \,a^{2} b^{3} d \,e^{3} x^{2} \ln \left (b x +a \right )-48 B \,a^{2} b^{3} e^{4} x^{3}+192 B a \,b^{4} d \,e^{3} x^{3}-72 B \,b^{5} d^{2} e^{2} x^{3}+48 A \,a^{3} b^{2} e^{4} x \ln \left (b x +a \right )+108 A \,a^{2} b^{3} e^{4} x^{2}-72 A a \,b^{4} d \,e^{3} x^{2}-36 A \,b^{5} d^{2} e^{2} x^{2}-240 B \,a^{4} b \,e^{4} x \ln \left (b x +a \right )+192 B \,a^{3} b^{2} d \,e^{3} x \ln \left (b x +a \right )-252 B \,a^{3} b^{2} e^{4} x^{2}+432 B \,a^{2} b^{3} d \,e^{3} x^{2}-108 B a \,b^{4} d^{2} e^{2} x^{2}-24 B \,b^{5} d^{3} e \,x^{2}+12 A \,a^{4} b \,e^{4} \ln \left (b x +a \right )+88 A \,a^{3} b^{2} e^{4} x -48 A \,a^{2} b^{3} d \,e^{3} x -24 A a \,b^{4} d^{2} e^{2} x -16 A \,b^{5} d^{3} e x -60 B \,a^{5} e^{4} \ln \left (b x +a \right )+48 B \,a^{4} b d \,e^{3} \ln \left (b x +a \right )-248 B \,a^{4} b \,e^{4} x +352 B \,a^{3} b^{2} d \,e^{3} x -72 B \,a^{2} b^{3} d^{2} e^{2} x -16 B a \,b^{4} d^{3} e x -4 B \,b^{5} d^{4} x +25 A \,a^{4} b \,e^{4}-12 A \,a^{3} b^{2} d \,e^{3}-6 A \,a^{2} b^{3} d^{2} e^{2}-4 A a \,b^{4} d^{3} e -3 A \,b^{5} d^{4}-77 B \,a^{5} e^{4}+100 B \,a^{4} b d \,e^{3}-18 B \,a^{3} b^{2} d^{2} e^{2}-4 B \,a^{2} b^{3} d^{3} e -B a \,b^{4} d^{4}\right ) \left (b x +a \right )}{12 \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} b^{6}} \]

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*(-4*A*a*b^4*d^3*e+192*B*a*b^4*d*e^3*x^3+100*B*a^4*b*d*e^3-18*B*a^3*b^2*d^2*e^2+192*B*ln(b*x+a)*x^3*a*b^4*
d*e^3-4*B*a^2*b^3*d^3*e-60*B*a^5*e^4*ln(b*x+a)-4*B*b^5*d^4*x+12*B*b^5*e^4*x^5+48*A*a*b^4*e^4*x^3-48*A*b^5*d*e^
3*x^3-48*B*a^2*b^3*e^4*x^3-12*A*a^3*b^2*d*e^3-72*B*a^2*b^3*d^2*e^2*x-48*A*a^2*b^3*d*e^3*x+352*B*a^3*b^2*d*e^3*
x+25*A*a^4*b*e^4-B*a*b^4*d^4-3*A*b^5*d^4-77*B*a^5*e^4-72*A*a*b^4*d*e^3*x^2+48*B*a^4*b*d*e^3*ln(b*x+a)+48*A*a^3
*b^2*e^4*x*ln(b*x+a)-240*B*a^4*b*e^4*x*ln(b*x+a)+72*A*a^2*b^3*e^4*x^2*ln(b*x+a)-360*B*a^3*b^2*e^4*x^2*ln(b*x+a
)-24*A*a*b^4*d^2*e^2*x-16*B*a*b^4*d^3*e*x-6*A*a^2*b^3*d^2*e^2-24*B*x^2*b^5*d^3*e+12*A*ln(b*x+a)*x^4*b^5*e^4+19
2*B*a^3*b^2*d*e^3*x*ln(b*x+a)+288*B*a^2*b^3*d*e^3*x^2*ln(b*x+a)-36*A*x^2*b^5*d^2*e^2-108*B*a*b^4*d^2*e^2*x^2+4
32*B*a^2*b^3*d*e^3*x^2-240*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+48*A*ln(b*x+a)*x^3*a*b^4*e^4+48*B*a*b^4*e^4*x^4-72*B*b^5*d^2*e^2*x^3+12*A*a^4*b*e^4*ln(b*x+a)-248*B*a^4*b*e
^4*x+108*A*a^2*b^3*e^4*x^2-252*B*a^3*b^2*e^4*x^2+88*A*a^3*b^2*e^4*x-16*A*b^5*d^3*e*x)*(b*x+a)/b^6/((b*x+a)^2)^
(5/2)

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maxima [B]  time = 0.95, size = 755, normalized size = 2.44 \[ \frac {1}{12} \, B e^{4} {\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 {1}{3} \, B d e^{3} {\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 {1}{12} \, A 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 {1}{2} \, B d^{2} e^{2} {\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 {1}{3} \, A d 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 {1}{12} \, B d^{4} {\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 {1}{3} \, A d^{3} 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 {1}{3} \, B d^{3} e {\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 {1}{2} \, A d^{2} 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 {A d^{4}}{4 \, b^{5} {\left (x + \frac {a}{b}\right )}^{4}} \]

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) + 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)) - 1/3*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) + 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^
4*(4/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*b^2) - 3*a/(b^6*(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^6*(x + a/b)^4)) - 1/3*B*d^3*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)) - 1/2*A*d^2*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^4/(b^5*(x + a/b)^4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(((A + B*x)*(d + e*x)^4)/(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 \[ \int \frac {\left (A + B x\right ) \left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \]

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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