\(\int \frac {(A+B x) (d+e x)^3}{(a+b x+c x^2)^{3/2}} \, dx\) [2473]

Optimal result

Integrand size = 27, antiderivative size = 325 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (15 b^3 B e^2-12 b^2 c e (3 B d+A e)-32 c^2 \left (A c d^2-3 a B d e-a A e^2\right )+4 b c \left (4 B c d^2+6 A c d e-13 a B e^2\right )-2 c e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2+5 b^2 e^2-4 c e (3 b d+a e)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \]

[Out]

3/8*e*(4*A*c*e*(-b*e+2*c*d)+B*(8*c^2*d^2+5*b^2*e^2-4*c*e*(a*e+3*b*d)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*
x+a)^(1/2))/c^(7/2)+2*(e*x+d)^2*(2*a*c*(A*e+B*d)-b*(A*c*d+B*a*e)-(b^2*B*e-b*c*(A*e+B*d)+2*c*(A*c*d-B*a*e))*x)/
c/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)-1/4*e*(15*b^3*B*e^2-12*b^2*c*e*(A*e+3*B*d)-32*c^2*(-A*a*e^2+A*c*d^2-3*B*a*d
*e)+4*b*c*(6*A*c*d*e-13*B*a*e^2+4*B*c*d^2)-2*c*e*(8*A*c^2*d+5*b^2*B*e-4*c*(A*b*e+3*B*a*e+B*b*d))*x)*(c*x^2+b*x
+a)^(1/2)/c^3/(-4*a*c+b^2)

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {832, 793, 635, 212} \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {3 e \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (B \left (-4 c e (a e+3 b d)+5 b^2 e^2+8 c^2 d^2\right )+4 A c e (2 c d-b e)\right )}{8 c^{7/2}}+\frac {2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c (3 a B e+A b e+b B d)+8 A c^2 d+5 b^2 B e\right )+4 b c \left (-13 a B e^2+6 A c d e+4 B c d^2\right )-32 c^2 \left (-a A e^2-3 a B d e+A c d^2\right )-12 b^2 c e (A e+3 B d)+15 b^3 B e^2\right )}{4 c^3 \left (b^2-4 a c\right )} \]

[In]

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

[Out]

(2*(d + e*x)^2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*x))/
(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (e*(15*b^3*B*e^2 - 12*b^2*c*e*(3*B*d + A*e) - 32*c^2*(A*c*d^2 - 3*a*
B*d*e - a*A*e^2) + 4*b*c*(4*B*c*d^2 + 6*A*c*d*e - 13*a*B*e^2) - 2*c*e*(8*A*c^2*d + 5*b^2*B*e - 4*c*(b*B*d + A*
b*e + 3*a*B*e))*x)*Sqrt[a + b*x + c*x^2])/(4*c^3*(b^2 - 4*a*c)) + (3*e*(4*A*c*e*(2*c*d - b*e) + B*(8*c^2*d^2 +
5*b^2*e^2 - 4*c*e*(3*b*d + a*e)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(7/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 793

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p +
3))), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(
a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 \int \frac {(d+e x) \left (\frac {1}{2} e \left (b^2 B d+4 A b c d-12 a B c d+4 a b B e-8 a A c e\right )+\frac {1}{2} e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )} \\ & = \frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (15 b^3 B e^2-12 b^2 c e (3 B d+A e)-32 c^2 \left (A c d^2-3 a B d e-a A e^2\right )+4 b c \left (4 B c d^2+6 A c d e-13 a B e^2\right )-2 c e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2+5 b^2 e^2-4 c e (3 b d+a e)\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^3} \\ & = \frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (15 b^3 B e^2-12 b^2 c e (3 B d+A e)-32 c^2 \left (A c d^2-3 a B d e-a A e^2\right )+4 b c \left (4 B c d^2+6 A c d e-13 a B e^2\right )-2 c e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2+5 b^2 e^2-4 c e (3 b d+a e)\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{4 c^3} \\ & = \frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (15 b^3 B e^2-12 b^2 c e (3 B d+A e)-32 c^2 \left (A c d^2-3 a B d e-a A e^2\right )+4 b c \left (4 B c d^2+6 A c d e-13 a B e^2\right )-2 c e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2+5 b^2 e^2-4 c e (3 b d+a e)\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.24 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.24 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-4 A c \left (3 b^3 e^3 x+b^2 e^2 (3 a e+c x (-6 d+e x))-2 b c \left (c d^2 (d-3 e x)+a e^2 (3 d+5 e x)\right )-4 c \left (2 a^2 e^3+c^2 d^3 x+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )\right )\right )+B \left (4 a^2 c e^2 (-13 b e+6 c (4 d+e x))+b x \left (-8 c^3 d^3+15 b^3 e^3+b^2 c e^2 (-36 d+5 e x)-2 b c^2 e \left (-12 d^2+6 d e x+e^2 x^2\right )\right )+a \left (15 b^3 e^3-2 b^2 c e^2 (18 d+31 e x)+4 b c^2 e \left (6 d^2+30 d e x-5 e^2 x^2\right )-8 c^3 \left (2 d^3+6 d^2 e x-6 d e^2 x^2-e^3 x^3\right )\right )\right )}{4 c^3 \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {3 e \left (4 A c e (-2 c d+b e)+B \left (-8 c^2 d^2-5 b^2 e^2+4 c e (3 b d+a e)\right )\right ) \log \left (c^3 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{8 c^{7/2}} \]

[In]

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

[Out]

(-4*A*c*(3*b^3*e^3*x + b^2*e^2*(3*a*e + c*x*(-6*d + e*x)) - 2*b*c*(c*d^2*(d - 3*e*x) + a*e^2*(3*d + 5*e*x)) -
4*c*(2*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + e^2*x^2))) + B*(4*a^2*c*e^2*(-13*b*e + 6*c*(4*d + e*x))
+ b*x*(-8*c^3*d^3 + 15*b^3*e^3 + b^2*c*e^2*(-36*d + 5*e*x) - 2*b*c^2*e*(-12*d^2 + 6*d*e*x + e^2*x^2)) + a*(15
*b^3*e^3 - 2*b^2*c*e^2*(18*d + 31*e*x) + 4*b*c^2*e*(6*d^2 + 30*d*e*x - 5*e^2*x^2) - 8*c^3*(2*d^3 + 6*d^2*e*x -
6*d*e^2*x^2 - e^3*x^3))))/(4*c^3*(-b^2 + 4*a*c)*Sqrt[a + x*(b + c*x)]) + (3*e*(4*A*c*e*(-2*c*d + b*e) + B*(-8
*c^2*d^2 - 5*b^2*e^2 + 4*c*e*(3*b*d + a*e)))*Log[c^3*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(7/2
))

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.70

method	result	size
risch	\(\frac {e^{2} \left (2 B c e x +4 A c e -7 B b e +12 B c d \right ) \sqrt {c \,x^{2}+b x +a}}{4 c^{3}}-\frac {-\frac {16 A \,c^{3} d^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 B \,e^{3} c \,a^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {14 B a \,b^{2} e^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 A a b c \,e^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {24 B a b c d \,e^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (12 A b \,c^{2} e^{3}-24 A \,c^{3} d \,e^{2}+12 B \,e^{3} a \,c^{2}-15 B \,e^{3} b^{2} c +36 B b \,c^{2} d \,e^{2}-24 B \,c^{3} d^{2} e \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (8 A a \,c^{2} e^{3}+4 A \,b^{2} c \,e^{3}-24 A \,c^{3} d^{2} e -4 B a b c \,e^{3}+24 B a \,c^{2} d \,e^{2}-7 B \,b^{3} e^{3}+12 B \,b^{2} c d \,e^{2}-8 B \,c^{3} d^{3}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{8 c^{3}}\)	\(554\)
default	\(\frac {2 A \,d^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+B \,e^{3} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )}{4 c}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+\left (A \,e^{3}+3 B d \,e^{2}\right ) \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+\left (3 A \,d^{2} e +B \,d^{3}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\)	\(778\)

[In]

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/4*e^2*(2*B*c*e*x+4*A*c*e-7*B*b*e+12*B*c*d)*(c*x^2+b*x+a)^(1/2)/c^3-1/8/c^3*(-16*A*c^3*d^3*(2*c*x+b)/(4*a*c-b
^2)/(c*x^2+b*x+a)^(1/2)+8*B*e^3*c*a^2*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-14*B*a*b^2*e^3*(2*c*x+b)/(4*a*
c-b^2)/(c*x^2+b*x+a)^(1/2)+8*A*a*b*c*e^3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+24*B*a*b*c*d*e^2*(2*c*x+b)/
(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+(12*A*b*c^2*e^3-24*A*c^3*d*e^2+12*B*a*c^2*e^3-15*B*b^2*c*e^3+36*B*b*c^2*d*e^2-
24*B*c^3*d^2*e)*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b
*x+a)^(1/2))+1/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+(8*A*a*c^2*e^3+4*A*b^2*c*e^3-24*A*c^3*d^2*
e-4*B*a*b*c*e^3+24*B*a*c^2*d*e^2-7*B*b^3*e^3+12*B*b^2*c*d*e^2-8*B*c^3*d^3)*(-1/c/(c*x^2+b*x+a)^(1/2)-b/c*(2*c*
x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (309) = 618\).

Time = 1.90 (sec) , antiderivative size = 1605, normalized size of antiderivative = 4.94 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/16*(3*(8*(B*a*b^2*c^2 - 4*B*a^2*c^3)*d^2*e - 4*(3*B*a*b^3*c + 8*A*a^2*c^3 - 2*(6*B*a^2*b + A*a*b^2)*c^2)*d
*e^2 + (5*B*a*b^4 + 16*(B*a^3 + A*a^2*b)*c^2 - 4*(6*B*a^2*b^2 + A*a*b^3)*c)*e^3 + (8*(B*b^2*c^3 - 4*B*a*c^4)*d
^2*e - 4*(3*B*b^3*c^2 + 8*A*a*c^4 - 2*(6*B*a*b + A*b^2)*c^3)*d*e^2 + (5*B*b^4*c + 16*(B*a^2 + A*a*b)*c^3 - 4*(
6*B*a*b^2 + A*b^3)*c^2)*e^3)*x^2 + (8*(B*b^3*c^2 - 4*B*a*b*c^3)*d^2*e - 4*(3*B*b^4*c + 8*A*a*b*c^3 - 2*(6*B*a*
b^2 + A*b^3)*c^2)*d*e^2 + (5*B*b^5 + 16*(B*a^2*b + A*a*b^2)*c^2 - 4*(6*B*a*b^3 + A*b^4)*c)*e^3)*x)*sqrt(c)*log
(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(8*(2*B*a - A*b)*c^4*d^
3 + 2*(B*b^2*c^3 - 4*B*a*c^4)*e^3*x^3 - 24*(B*a*b*c^3 - 2*A*a*c^4)*d^2*e + 12*(3*B*a*b^2*c^2 - 2*(4*B*a^2 + A*
a*b)*c^3)*d*e^2 - (15*B*a*b^3*c + 32*A*a^2*c^3 - 4*(13*B*a^2*b + 3*A*a*b^2)*c^2)*e^3 + (12*(B*b^2*c^3 - 4*B*a*
c^4)*d*e^2 - (5*B*b^3*c^2 + 16*A*a*c^4 - 4*(5*B*a*b + A*b^2)*c^3)*e^3)*x^2 + (8*(B*b*c^4 - 2*A*c^5)*d^3 - 24*(
B*b^2*c^3 - (2*B*a + A*b)*c^4)*d^2*e + 12*(3*B*b^3*c^2 + 4*A*a*c^4 - 2*(5*B*a*b + A*b^2)*c^3)*d*e^2 - (15*B*b^
4*c + 8*(3*B*a^2 + 5*A*a*b)*c^3 - 2*(31*B*a*b^2 + 6*A*b^3)*c^2)*e^3)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^4 - 4*
a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4 - 4*a*b*c^5)*x), -1/8*(3*(8*(B*a*b^2*c^2 - 4*B*a^2*c^3)*d^2*e - 4
*(3*B*a*b^3*c + 8*A*a^2*c^3 - 2*(6*B*a^2*b + A*a*b^2)*c^2)*d*e^2 + (5*B*a*b^4 + 16*(B*a^3 + A*a^2*b)*c^2 - 4*(
6*B*a^2*b^2 + A*a*b^3)*c)*e^3 + (8*(B*b^2*c^3 - 4*B*a*c^4)*d^2*e - 4*(3*B*b^3*c^2 + 8*A*a*c^4 - 2*(6*B*a*b + A
*b^2)*c^3)*d*e^2 + (5*B*b^4*c + 16*(B*a^2 + A*a*b)*c^3 - 4*(6*B*a*b^2 + A*b^3)*c^2)*e^3)*x^2 + (8*(B*b^3*c^2 -
4*B*a*b*c^3)*d^2*e - 4*(3*B*b^4*c + 8*A*a*b*c^3 - 2*(6*B*a*b^2 + A*b^3)*c^2)*d*e^2 + (5*B*b^5 + 16*(B*a^2*b +
A*a*b^2)*c^2 - 4*(6*B*a*b^3 + A*b^4)*c)*e^3)*x)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c
)/(c^2*x^2 + b*c*x + a*c)) - 2*(8*(2*B*a - A*b)*c^4*d^3 + 2*(B*b^2*c^3 - 4*B*a*c^4)*e^3*x^3 - 24*(B*a*b*c^3 -
2*A*a*c^4)*d^2*e + 12*(3*B*a*b^2*c^2 - 2*(4*B*a^2 + A*a*b)*c^3)*d*e^2 - (15*B*a*b^3*c + 32*A*a^2*c^3 - 4*(13*B
*a^2*b + 3*A*a*b^2)*c^2)*e^3 + (12*(B*b^2*c^3 - 4*B*a*c^4)*d*e^2 - (5*B*b^3*c^2 + 16*A*a*c^4 - 4*(5*B*a*b + A*
b^2)*c^3)*e^3)*x^2 + (8*(B*b*c^4 - 2*A*c^5)*d^3 - 24*(B*b^2*c^3 - (2*B*a + A*b)*c^4)*d^2*e + 12*(3*B*b^3*c^2 +
4*A*a*c^4 - 2*(5*B*a*b + A*b^2)*c^3)*d*e^2 - (15*B*b^4*c + 8*(3*B*a^2 + 5*A*a*b)*c^3 - 2*(31*B*a*b^2 + 6*A*b^
3)*c^2)*e^3)*x)*sqrt(c*x^2 + b*x + a))/(a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^2 + (b^3*c^4 - 4*a*b*c^5
)*x)]

Sympy [F]

\[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{3}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x+a)**(3/2),x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, {\left (B b^{2} c^{2} e^{3} - 4 \, B a c^{3} e^{3}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac {12 \, B b^{2} c^{2} d e^{2} - 48 \, B a c^{3} d e^{2} - 5 \, B b^{3} c e^{3} + 20 \, B a b c^{2} e^{3} + 4 \, A b^{2} c^{2} e^{3} - 16 \, A a c^{3} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x + \frac {8 \, B b c^{3} d^{3} - 16 \, A c^{4} d^{3} - 24 \, B b^{2} c^{2} d^{2} e + 48 \, B a c^{3} d^{2} e + 24 \, A b c^{3} d^{2} e + 36 \, B b^{3} c d e^{2} - 120 \, B a b c^{2} d e^{2} - 24 \, A b^{2} c^{2} d e^{2} + 48 \, A a c^{3} d e^{2} - 15 \, B b^{4} e^{3} + 62 \, B a b^{2} c e^{3} + 12 \, A b^{3} c e^{3} - 24 \, B a^{2} c^{2} e^{3} - 40 \, A a b c^{2} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x + \frac {16 \, B a c^{3} d^{3} - 8 \, A b c^{3} d^{3} - 24 \, B a b c^{2} d^{2} e + 48 \, A a c^{3} d^{2} e + 36 \, B a b^{2} c d e^{2} - 96 \, B a^{2} c^{2} d e^{2} - 24 \, A a b c^{2} d e^{2} - 15 \, B a b^{3} e^{3} + 52 \, B a^{2} b c e^{3} + 12 \, A a b^{2} c e^{3} - 32 \, A a^{2} c^{2} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (8 \, B c^{2} d^{2} e - 12 \, B b c d e^{2} + 8 \, A c^{2} d e^{2} + 5 \, B b^{2} e^{3} - 4 \, B a c e^{3} - 4 \, A b c e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {7}{2}}} \]

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/4*(((2*(B*b^2*c^2*e^3 - 4*B*a*c^3*e^3)*x/(b^2*c^3 - 4*a*c^4) + (12*B*b^2*c^2*d*e^2 - 48*B*a*c^3*d*e^2 - 5*B*
b^3*c*e^3 + 20*B*a*b*c^2*e^3 + 4*A*b^2*c^2*e^3 - 16*A*a*c^3*e^3)/(b^2*c^3 - 4*a*c^4))*x + (8*B*b*c^3*d^3 - 16*
A*c^4*d^3 - 24*B*b^2*c^2*d^2*e + 48*B*a*c^3*d^2*e + 24*A*b*c^3*d^2*e + 36*B*b^3*c*d*e^2 - 120*B*a*b*c^2*d*e^2
- 24*A*b^2*c^2*d*e^2 + 48*A*a*c^3*d*e^2 - 15*B*b^4*e^3 + 62*B*a*b^2*c*e^3 + 12*A*b^3*c*e^3 - 24*B*a^2*c^2*e^3
- 40*A*a*b*c^2*e^3)/(b^2*c^3 - 4*a*c^4))*x + (16*B*a*c^3*d^3 - 8*A*b*c^3*d^3 - 24*B*a*b*c^2*d^2*e + 48*A*a*c^3
*d^2*e + 36*B*a*b^2*c*d*e^2 - 96*B*a^2*c^2*d*e^2 - 24*A*a*b*c^2*d*e^2 - 15*B*a*b^3*e^3 + 52*B*a^2*b*c*e^3 + 12
*A*a*b^2*c*e^3 - 32*A*a^2*c^2*e^3)/(b^2*c^3 - 4*a*c^4))/sqrt(c*x^2 + b*x + a) - 3/8*(8*B*c^2*d^2*e - 12*B*b*c*
d*e^2 + 8*A*c^2*d*e^2 + 5*B*b^2*e^3 - 4*B*a*c*e^3 - 4*A*b*c*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
*sqrt(c) + b))/c^(7/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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