3.1767 \(\int \frac {(A+B x) (d+e x)^4}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=329 \[ -\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e (a+b x) (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 x (a+b x) \left (6 a^2 B e^2-3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 x^2 (a+b x) (-3 a B e+A b e+4 b B d)}{2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^4 x^3 (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]

[Out]

-(-a*e+b*d)^3*(4*A*b*e-5*B*a*e+B*b*d)/b^6/((b*x+a)^2)^(1/2)-1/2*(A*b-B*a)*(-a*e+b*d)^4/b^6/(b*x+a)/((b*x+a)^2)
^(1/2)+e^2*(6*a^2*B*e^2-3*a*b*e*(A*e+4*B*d)+2*b^2*d*(2*A*e+3*B*d))*x*(b*x+a)/b^5/((b*x+a)^2)^(1/2)+1/2*e^3*(A*
b*e-3*B*a*e+4*B*b*d)*x^2*(b*x+a)/b^4/((b*x+a)^2)^(1/2)+1/3*B*e^4*x^3*(b*x+a)/b^3/((b*x+a)^2)^(1/2)+2*e*(-a*e+b
*d)^2*(3*A*b*e-5*B*a*e+2*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 = 0.37, antiderivative size = 329, 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} \[ \frac {e^3 x^2 (a+b x) (-3 a B e+A b e+4 b B d)}{2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 x (a+b x) \left (6 a^2 B e^2-3 a b e (A e+4 B d)+2 b^2 d (2 A e+3 B d)\right )}{b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(b d-a e)^3 (-5 a B e+4 A b e+b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e (a+b x) (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^4 x^3 (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

-(((b*d - a*e)^3*(b*B*d + 4*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)/(2*b^6*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (e^2*(6*a^2*B*e^2 - 3*a*b*e*(4*B*d + A*e) + 2*b^2*d*(3*B*
d + 2*A*e))*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 - 3*a*B*e)*x^2*(a + b*x))
/(2*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (B*e^4*x^3*(a + b*x))/(3*b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (2*e*(b
*d - a*e)^2*(2*b*B*d + 3*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 )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^4}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \left (\frac {e^2 \left (6 a^2 B e^2-3 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right )}{b^8}+\frac {e^3 (4 b B d+A b e-3 a B e) x}{b^7}+\frac {B e^4 x^2}{b^6}+\frac {(A b-a B) (b d-a e)^4}{b^8 (a+b x)^3}+\frac {(b d-a e)^3 (b B d+4 A b e-5 a B e)}{b^8 (a+b x)^2}+\frac {2 e (b d-a e)^2 (2 b B d+3 A b e-5 a B e)}{b^8 (a+b x)}\right ) \, dx}{\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)}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (b d-a e)^4}{2 b^6 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 \left (6 a^2 B e^2-3 a b e (4 B d+A e)+2 b^2 d (3 B d+2 A e)\right ) 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-3 a B e) x^2 (a+b x)}{2 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {B e^4 x^3 (a+b x)}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 e (b d-a e)^2 (2 b B d+3 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 = 0.23, size = 373, normalized size = 1.13 \[ \frac {-3 A b \left (-7 a^4 e^4-2 a^3 b e^3 (e x-10 d)+a^2 b^2 e^2 \left (-18 d^2+16 d e x+11 e^2 x^2\right )+4 a b^3 e \left (d^3-6 d^2 e x-4 d e^2 x^2+e^3 x^3\right )+b^4 \left (d^4+8 d^3 e x-8 d e^3 x^3-e^4 x^4\right )\right )+B \left (-27 a^5 e^4+6 a^4 b e^3 (14 d+e x)+3 a^3 b^2 e^2 \left (-30 d^2+8 d e x+21 e^2 x^2\right )+4 a^2 b^3 e \left (9 d^3-18 d^2 e x-33 d e^2 x^2+5 e^3 x^3\right )+a b^4 \left (-3 d^4+48 d^3 e x+72 d^2 e^2 x^2-48 d e^3 x^3-5 e^4 x^4\right )+2 b^5 x \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )\right )+12 e (a+b x)^2 (b d-a e)^2 \log (a+b x) (-5 a B e+3 A b e+2 b B d)}{6 b^6 (a+b x) \sqrt {(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(-3*A*b*(-7*a^4*e^4 - 2*a^3*b*e^3*(-10*d + e*x) + a^2*b^2*e^2*(-18*d^2 + 16*d*e*x + 11*e^2*x^2) + 4*a*b^3*e*(d
^3 - 6*d^2*e*x - 4*d*e^2*x^2 + e^3*x^3) + b^4*(d^4 + 8*d^3*e*x - 8*d*e^3*x^3 - e^4*x^4)) + B*(-27*a^5*e^4 + 6*
a^4*b*e^3*(14*d + e*x) + 3*a^3*b^2*e^2*(-30*d^2 + 8*d*e*x + 21*e^2*x^2) + 4*a^2*b^3*e*(9*d^3 - 18*d^2*e*x - 33
*d*e^2*x^2 + 5*e^3*x^3) + a*b^4*(-3*d^4 + 48*d^3*e*x + 72*d^2*e^2*x^2 - 48*d*e^3*x^3 - 5*e^4*x^4) + 2*b^5*x*(-
3*d^4 + 18*d^2*e^2*x^2 + 6*d*e^3*x^3 + e^4*x^4)) + 12*e*(b*d - a*e)^2*(2*b*B*d + 3*A*b*e - 5*a*B*e)*(a + b*x)^
2*Log[a + b*x])/(6*b^6*(a + b*x)*Sqrt[(a + b*x)^2])

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fricas [B]  time = 0.82, size = 668, normalized size = 2.03 \[ \frac {2 \, B b^{5} e^{4} x^{5} - 3 \, {\left (B a b^{4} + A b^{5}\right )} d^{4} + 12 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e - 18 \, {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{2} + 12 \, {\left (7 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} d e^{3} - 3 \, {\left (9 \, B a^{5} - 7 \, A a^{4} b\right )} e^{4} + {\left (12 \, B b^{5} d e^{3} - {\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} e^{4}\right )} x^{4} + 4 \, {\left (9 \, B b^{5} d^{2} e^{2} - 6 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d e^{3} + {\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 3 \, {\left (24 \, B a b^{4} d^{2} e^{2} - 4 \, {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} d e^{3} + {\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} - 6 \, {\left (B b^{5} d^{4} - 4 \, {\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 12 \, {\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} - 4 \, {\left (B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} - {\left (B a^{4} b + A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \, {\left (2 \, B a^{2} b^{3} d^{3} e - 3 \, {\left (3 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{4} b - A a^{3} b^{2}\right )} d e^{3} - {\left (5 \, B a^{5} - 3 \, A a^{4} b\right )} e^{4} + {\left (2 \, B b^{5} d^{3} e - 3 \, {\left (3 \, B a b^{4} - A b^{5}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d e^{3} - {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 2 \, {\left (2 \, B a b^{4} d^{3} e - 3 \, {\left (3 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 6 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{3} - {\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]

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)^(3/2),x, algorithm="fricas")

[Out]

1/6*(2*B*b^5*e^4*x^5 - 3*(B*a*b^4 + A*b^5)*d^4 + 12*(3*B*a^2*b^3 - A*a*b^4)*d^3*e - 18*(5*B*a^3*b^2 - 3*A*a^2*
b^3)*d^2*e^2 + 12*(7*B*a^4*b - 5*A*a^3*b^2)*d*e^3 - 3*(9*B*a^5 - 7*A*a^4*b)*e^4 + (12*B*b^5*d*e^3 - (5*B*a*b^4
- 3*A*b^5)*e^4)*x^4 + 4*(9*B*b^5*d^2*e^2 - 6*(2*B*a*b^4 - A*b^5)*d*e^3 + (5*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 +
3*(24*B*a*b^4*d^2*e^2 - 4*(11*B*a^2*b^3 - 4*A*a*b^4)*d*e^3 + (21*B*a^3*b^2 - 11*A*a^2*b^3)*e^4)*x^2 - 6*(B*b^
5*d^4 - 4*(2*B*a*b^4 - A*b^5)*d^3*e + 12*(B*a^2*b^3 - A*a*b^4)*d^2*e^2 - 4*(B*a^3*b^2 - 2*A*a^2*b^3)*d*e^3 - (
B*a^4*b + A*a^3*b^2)*e^4)*x + 12*(2*B*a^2*b^3*d^3*e - 3*(3*B*a^3*b^2 - A*a^2*b^3)*d^2*e^2 + 6*(2*B*a^4*b - A*a
^3*b^2)*d*e^3 - (5*B*a^5 - 3*A*a^4*b)*e^4 + (2*B*b^5*d^3*e - 3*(3*B*a*b^4 - A*b^5)*d^2*e^2 + 6*(2*B*a^2*b^3 -
A*a*b^4)*d*e^3 - (5*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 + 2*(2*B*a*b^4*d^3*e - 3*(3*B*a^2*b^3 - A*a*b^4)*d^2*e^2
+ 6*(2*B*a^3*b^2 - A*a^2*b^3)*d*e^3 - (5*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x +
a^2*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)^(3/2),x, algorithm="giac")

[Out]

sage0*x

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maple [B]  time = 0.08, size = 858, normalized size = 2.61 \[ \frac {\left (2 B \,b^{5} e^{4} x^{5}+3 A \,b^{5} e^{4} x^{4}-5 B a \,b^{4} e^{4} x^{4}+12 B \,b^{5} d \,e^{3} x^{4}+36 A \,a^{2} b^{3} e^{4} x^{2} \ln \left (b x +a \right )-72 A a \,b^{4} d \,e^{3} x^{2} \ln \left (b x +a \right )-12 A a \,b^{4} e^{4} x^{3}+36 A \,b^{5} d^{2} e^{2} x^{2} \ln \left (b x +a \right )+24 A \,b^{5} d \,e^{3} x^{3}-60 B \,a^{3} b^{2} e^{4} x^{2} \ln \left (b x +a \right )+144 B \,a^{2} b^{3} d \,e^{3} x^{2} \ln \left (b x +a \right )+20 B \,a^{2} b^{3} e^{4} x^{3}-108 B a \,b^{4} d^{2} e^{2} x^{2} \ln \left (b x +a \right )-48 B a \,b^{4} d \,e^{3} x^{3}+24 B \,b^{5} d^{3} e \,x^{2} \ln \left (b x +a \right )+36 B \,b^{5} d^{2} e^{2} x^{3}+72 A \,a^{3} b^{2} e^{4} x \ln \left (b x +a \right )-144 A \,a^{2} b^{3} d \,e^{3} x \ln \left (b x +a \right )-33 A \,a^{2} b^{3} e^{4} x^{2}+72 A a \,b^{4} d^{2} e^{2} x \ln \left (b x +a \right )+48 A a \,b^{4} d \,e^{3} x^{2}-120 B \,a^{4} b \,e^{4} x \ln \left (b x +a \right )+288 B \,a^{3} b^{2} d \,e^{3} x \ln \left (b x +a \right )+63 B \,a^{3} b^{2} e^{4} x^{2}-216 B \,a^{2} b^{3} d^{2} e^{2} x \ln \left (b x +a \right )-132 B \,a^{2} b^{3} d \,e^{3} x^{2}+48 B a \,b^{4} d^{3} e x \ln \left (b x +a \right )+72 B a \,b^{4} d^{2} e^{2} x^{2}+36 A \,a^{4} b \,e^{4} \ln \left (b x +a \right )-72 A \,a^{3} b^{2} d \,e^{3} \ln \left (b x +a \right )+6 A \,a^{3} b^{2} e^{4} x +36 A \,a^{2} b^{3} d^{2} e^{2} \ln \left (b x +a \right )-48 A \,a^{2} b^{3} d \,e^{3} x +72 A a \,b^{4} d^{2} e^{2} x -24 A \,b^{5} d^{3} e x -60 B \,a^{5} e^{4} \ln \left (b x +a \right )+144 B \,a^{4} b d \,e^{3} \ln \left (b x +a \right )+6 B \,a^{4} b \,e^{4} x -108 B \,a^{3} b^{2} d^{2} e^{2} \ln \left (b x +a \right )+24 B \,a^{3} b^{2} d \,e^{3} x +24 B \,a^{2} b^{3} d^{3} e \ln \left (b x +a \right )-72 B \,a^{2} b^{3} d^{2} e^{2} x +48 B a \,b^{4} d^{3} e x -6 B \,b^{5} d^{4} x +21 A \,a^{4} b \,e^{4}-60 A \,a^{3} b^{2} d \,e^{3}+54 A \,a^{2} b^{3} d^{2} e^{2}-12 A a \,b^{4} d^{3} e -3 A \,b^{5} d^{4}-27 B \,a^{5} e^{4}+84 B \,a^{4} b d \,e^{3}-90 B \,a^{3} b^{2} d^{2} e^{2}+36 B \,a^{2} b^{3} d^{3} e -3 B a \,b^{4} d^{4}\right ) \left (b x +a \right )}{6 \left (\left (b x +a \right )^{2}\right )^{\frac {3}{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)^(3/2),x)

[Out]

1/6*(-12*A*a*b^4*d^3*e-48*B*x^3*a*b^4*d*e^3+84*B*a^4*b*d*e^3-90*B*a^3*b^2*d^2*e^2+36*b^3*B*a^2*d^3*e-60*B*ln(b
*x+a)*a^5*e^4-6*B*x*b^5*d^4+2*B*x^5*b^5*e^4+3*A*x^4*b^5*e^4-12*A*x^3*a*b^4*e^4+24*A*x^3*b^5*d*e^3+20*B*x^3*a^2
*b^3*e^4-60*A*a^3*b^2*d*e^3-108*B*ln(b*x+a)*a^3*b^2*d^2*e^2+24*B*ln(b*x+a)*a^2*b^3*d^3*e-72*B*x*a^2*b^3*d^2*e^
2-48*A*x*a^2*b^3*d*e^3+24*B*x*a^3*b^2*d*e^3+21*A*a^4*b*e^4-3*B*a*b^4*d^4-3*A*b^5*d^4-27*B*a^5*e^4+48*A*x^2*a*b
^4*d*e^3+144*B*ln(b*x+a)*a^4*b*d*e^3+36*A*ln(b*x+a)*a^2*b^3*d^2*e^2-72*A*ln(b*x+a)*a^3*b^2*d*e^3+24*B*ln(b*x+a
)*x^2*b^5*d^3*e+72*A*ln(b*x+a)*x*a^3*b^2*e^4-120*B*ln(b*x+a)*x*a^4*b*e^4+36*A*ln(b*x+a)*x^2*a^2*b^3*e^4+36*A*l
n(b*x+a)*x^2*b^5*d^2*e^2-60*B*ln(b*x+a)*x^2*a^3*b^2*e^4+72*A*x*a*b^4*d^2*e^2+48*B*x*a*b^4*d^3*e+54*A*a^2*b^3*d
^2*e^2-108*B*ln(b*x+a)*x^2*a*b^4*d^2*e^2-144*A*ln(b*x+a)*x*a^2*b^3*d*e^3-72*A*ln(b*x+a)*x^2*a*b^4*d*e^3+288*B*
ln(b*x+a)*x*a^3*b^2*d*e^3-216*B*ln(b*x+a)*x*a^2*b^3*d^2*e^2+72*A*ln(b*x+a)*x*a*b^4*d^2*e^2+48*B*ln(b*x+a)*x*a*
b^4*d^3*e+144*B*ln(b*x+a)*x^2*a^2*b^3*d*e^3+72*B*x^2*a*b^4*d^2*e^2-132*B*x^2*a^2*b^3*d*e^3-5*B*x^4*a*b^4*e^4+3
6*B*x^3*b^5*d^2*e^2+36*A*ln(b*x+a)*a^4*b*e^4+12*B*x^4*b^5*d*e^3+6*B*x*a^4*b*e^4-33*A*x^2*a^2*b^3*e^4+63*B*x^2*
a^3*b^2*e^4+6*A*x*a^3*b^2*e^4-24*A*x*b^5*d^3*e)*(b*x+a)/b^6/((b*x+a)^2)^(3/2)

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maxima [B]  time = 0.53, size = 761, normalized size = 2.31 \[ \frac {B e^{4} x^{4}}{3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {7 \, B a e^{4} x^{3}}{6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {9 \, B a^{2} e^{4} x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {10 \, B a^{3} e^{4} \log \left (x + \frac {a}{b}\right )}{b^{6}} + \frac {9 \, B a^{4} e^{4}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{6}} + \frac {{\left (4 \, B d e^{3} + A e^{4}\right )} x^{3}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {20 \, B a^{4} e^{4} x}{b^{7} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {5 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a x^{2}}{2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{3}} + \frac {2 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} - \frac {A d^{4}}{2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {39 \, B a^{5} e^{4}}{2 \, b^{8} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {6 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a^{2} \log \left (x + \frac {a}{b}\right )}{b^{5}} - \frac {6 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a \log \left (x + \frac {a}{b}\right )}{b^{4}} + \frac {2 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} \log \left (x + \frac {a}{b}\right )}{b^{3}} - \frac {5 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a^{3}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{5}} + \frac {4 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{4}} - \frac {B d^{4} + 4 \, A d^{3} e}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} b^{2}} + \frac {12 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a^{3} x}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {12 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{2} x}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {4 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} a x}{b^{4} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {23 \, {\left (4 \, B d e^{3} + A e^{4}\right )} a^{4}}{2 \, b^{7} {\left (x + \frac {a}{b}\right )}^{2}} - \frac {11 \, {\left (3 \, B d^{2} e^{2} + 2 \, A d e^{3}\right )} a^{3}}{b^{6} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {3 \, {\left (2 \, B d^{3} e + 3 \, A d^{2} e^{2}\right )} a^{2}}{b^{5} {\left (x + \frac {a}{b}\right )}^{2}} + \frac {{\left (B d^{4} + 4 \, A d^{3} e\right )} a}{2 \, b^{4} {\left (x + \frac {a}{b}\right )}^{2}} \]

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)^(3/2),x, algorithm="maxima")

[Out]

1/3*B*e^4*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 7/6*B*a*e^4*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^3) + 9/2*
B*a^2*e^4*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4) - 10*B*a^3*e^4*log(x + a/b)/b^6 + 9*B*a^4*e^4/(sqrt(b^2*x^2
+ 2*a*b*x + a^2)*b^6) + 1/2*(4*B*d*e^3 + A*e^4)*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 20*B*a^4*e^4*x/(b^7*
(x + a/b)^2) - 5/2*(4*B*d*e^3 + A*e^4)*a*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^3) + 2*(3*B*d^2*e^2 + 2*A*d*e^3)
*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) - 1/2*A*d^4/(b^3*(x + a/b)^2) - 39/2*B*a^5*e^4/(b^8*(x + a/b)^2) + 6*
(4*B*d*e^3 + A*e^4)*a^2*log(x + a/b)/b^5 - 6*(3*B*d^2*e^2 + 2*A*d*e^3)*a*log(x + a/b)/b^4 + 2*(2*B*d^3*e + 3*A
*d^2*e^2)*log(x + a/b)/b^3 - 5*(4*B*d*e^3 + A*e^4)*a^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^5) + 4*(3*B*d^2*e^2 +
2*A*d*e^3)*a^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^4) - (B*d^4 + 4*A*d^3*e)/(sqrt(b^2*x^2 + 2*a*b*x + a^2)*b^2) +
12*(4*B*d*e^3 + A*e^4)*a^3*x/(b^6*(x + a/b)^2) - 12*(3*B*d^2*e^2 + 2*A*d*e^3)*a^2*x/(b^5*(x + a/b)^2) + 4*(2*
B*d^3*e + 3*A*d^2*e^2)*a*x/(b^4*(x + a/b)^2) + 23/2*(4*B*d*e^3 + A*e^4)*a^4/(b^7*(x + a/b)^2) - 11*(3*B*d^2*e^
2 + 2*A*d*e^3)*a^3/(b^6*(x + a/b)^2) + 3*(2*B*d^3*e + 3*A*d^2*e^2)*a^2/(b^5*(x + a/b)^2) + 1/2*(B*d^4 + 4*A*d^
3*e)*a/(b^4*(x + a/b)^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^4}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]

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)^(3/2),x)

[Out]

int(((A + B*x)*(d + e*x)^4)/(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B x\right ) \left (d + e x\right )^{4}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral((A + B*x)*(d + e*x)**4/((a + b*x)**2)**(3/2), x)

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