3.23.24 \(\int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx\)

Optimal. Leaf size=407 \[ \frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (B \left (-4 c e (9 a e+16 b d)+35 b^2 e^2+24 c^2 d^2\right )+40 A c e (2 c d-b e)\right )+8 A c e \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+64 c^2 d^2\right )+B \left (-8 c^2 d e (48 a e+47 b d)+20 b c e^2 (11 a e+18 b d)-105 b^3 e^3+96 c^3 d^3\right )\right )}{192 c^4}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right )}{128 c^{9/2}}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c} \]

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Rubi [A]  time = 0.68, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {832, 779, 621, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (B \left (-4 c e (9 a e+16 b d)+35 b^2 e^2+24 c^2 d^2\right )+40 A c e (2 c d-b e)\right )+8 A c e \left (-2 c e (8 a e+27 b d)+15 b^2 e^2+64 c^2 d^2\right )+B \left (-8 c^2 d e (48 a e+47 b d)+20 b c e^2 (11 a e+18 b d)-105 b^3 e^3+96 c^3 d^3\right )\right )}{192 c^4}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right )}{128 c^{9/2}}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*B*c*d - 7*b*B*e + 8*A*c*e)*(d + e*x)^2*Sqrt[a + b*x + c*x^2])/(24*c^2) + (B*(d + e*x)^3*Sqrt[a + b*x + c*x
^2])/(4*c) + ((8*A*c*e*(64*c^2*d^2 + 15*b^2*e^2 - 2*c*e*(27*b*d + 8*a*e)) + B*(96*c^3*d^3 - 105*b^3*e^3 + 20*b
*c*e^2*(18*b*d + 11*a*e) - 8*c^2*d*e*(47*b*d + 48*a*e)) + 2*c*e*(40*A*c*e*(2*c*d - b*e) + B*(24*c^2*d^2 + 35*b
^2*e^2 - 4*c*e*(16*b*d + 9*a*e)))*x)*Sqrt[a + b*x + c*x^2])/(192*c^4) + ((35*b^4*B*e^3 - 40*b^3*c*e^2*(3*B*d +
A*e) + 24*b^2*c*e*(6*B*c*d^2 + 6*A*c*d*e - 5*a*B*e^2) - 32*b*c^2*(2*B*c*d^3 + 6*A*c*d^2*e - 9*a*B*d*e^2 - 3*a
*A*e^3) + 16*c^2*(4*A*c*d*(2*c*d^2 - 3*a*e^2) - 3*a*B*e*(4*c*d^2 - a*e^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqr
t[a + b*x + c*x^2])])/(128*c^(9/2))

Rule 206

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

Rule 621

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 779

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[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx &=\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\int \frac {(d+e x)^2 \left (\frac {1}{2} (-b B d+8 A c d-6 a B e)+\frac {1}{2} (6 B c d-7 b B e+8 A c e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{4 c}\\ &=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\int \frac {(d+e x) \left (\frac {1}{4} \left (7 b^2 B d e+4 c \left (12 A c d^2-15 a B d e-8 a A e^2\right )-4 b \left (3 B c d^2+2 A c d e-7 a B e^2\right )\right )+\frac {1}{4} \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2+35 b^2 e^2-4 c e (16 b d+9 a e)\right )\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{12 c^2}\\ &=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2+15 b^2 e^2-2 c e (27 b d+8 a e)\right )+B \left (96 c^3 d^3-105 b^3 e^3+20 b c e^2 (18 b d+11 a e)-8 c^2 d e (47 b d+48 a e)\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2+35 b^2 e^2-4 c e (16 b d+9 a e)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (35 b^4 B e^3-40 b^3 c e^2 (3 B d+A e)+24 b^2 c e \left (6 B c d^2+6 A c d e-5 a B e^2\right )-32 b c^2 \left (2 B c d^3+6 A c d^2 e-9 a B d e^2-3 a A e^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{128 c^4}\\ &=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2+15 b^2 e^2-2 c e (27 b d+8 a e)\right )+B \left (96 c^3 d^3-105 b^3 e^3+20 b c e^2 (18 b d+11 a e)-8 c^2 d e (47 b d+48 a e)\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2+35 b^2 e^2-4 c e (16 b d+9 a e)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (35 b^4 B e^3-40 b^3 c e^2 (3 B d+A e)+24 b^2 c e \left (6 B c d^2+6 A c d e-5 a B e^2\right )-32 b c^2 \left (2 B c d^3+6 A c d^2 e-9 a B d e^2-3 a A e^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{64 c^4}\\ &=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {a+b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {a+b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2+15 b^2 e^2-2 c e (27 b d+8 a e)\right )+B \left (96 c^3 d^3-105 b^3 e^3+20 b c e^2 (18 b d+11 a e)-8 c^2 d e (47 b d+48 a e)\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2+35 b^2 e^2-4 c e (16 b d+9 a e)\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{192 c^4}+\frac {\left (35 b^4 B e^3-40 b^3 c e^2 (3 B d+A e)+24 b^2 c e \left (6 B c d^2+6 A c d e-5 a B e^2\right )-32 b c^2 \left (2 B c d^3+6 A c d^2 e-9 a B d e^2-3 a A e^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )-3 a B e \left (4 c d^2-a e^2\right )\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.53, size = 357, normalized size = 0.88 \begin {gather*} \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (8 A c e \left (-2 c e (8 a e+27 b d+5 b e x)+15 b^2 e^2+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+B \left (-8 c^2 e \left (3 a e (16 d+3 e x)+b \left (54 d^2+30 d e x+7 e^2 x^2\right )\right )+10 b c e^2 (22 a e+36 b d+7 b e x)-105 b^3 e^3+48 c^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )\right )+3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right ) \left (24 b^2 c e \left (-5 a B e^2+6 A c d e+6 B c d^2\right )-32 b c^2 \left (-3 a A e^3-9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+16 c^2 \left (4 A c d \left (2 c d^2-3 a e^2\right )+3 a B e \left (a e^2-4 c d^2\right )\right )-40 b^3 c e^2 (A e+3 B d)+35 b^4 B e^3\right )}{384 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(8*A*c*e*(15*b^2*e^2 - 2*c*e*(27*b*d + 8*a*e + 5*b*e*x) + 4*c^2*(18*d^2 + 9*d
*e*x + 2*e^2*x^2)) + B*(-105*b^3*e^3 + 10*b*c*e^2*(36*b*d + 22*a*e + 7*b*e*x) + 48*c^3*(4*d^3 + 6*d^2*e*x + 4*
d*e^2*x^2 + e^3*x^3) - 8*c^2*e*(3*a*e*(16*d + 3*e*x) + b*(54*d^2 + 30*d*e*x + 7*e^2*x^2)))) + 3*(35*b^4*B*e^3
- 40*b^3*c*e^2*(3*B*d + A*e) + 24*b^2*c*e*(6*B*c*d^2 + 6*A*c*d*e - 5*a*B*e^2) - 32*b*c^2*(2*B*c*d^3 + 6*A*c*d^
2*e - 9*a*B*d*e^2 - 3*a*A*e^3) + 16*c^2*(4*A*c*d*(2*c*d^2 - 3*a*e^2) + 3*a*B*e*(-4*c*d^2 + a*e^2)))*ArcTanh[(b
+ 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(384*c^(9/2))

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IntegrateAlgebraic [A]  time = 1.77, size = 433, normalized size = 1.06 \begin {gather*} \frac {\log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right ) \left (-48 a^2 B c^2 e^3-96 a A b c^2 e^3+192 a A c^3 d e^2+120 a b^2 B c e^3-288 a b B c^2 d e^2+192 a B c^3 d^2 e+40 A b^3 c e^3-144 A b^2 c^2 d e^2+192 A b c^3 d^2 e-128 A c^4 d^3-35 b^4 B e^3+120 b^3 B c d e^2-144 b^2 B c^2 d^2 e+64 b B c^3 d^3\right )}{128 c^{9/2}}+\frac {\sqrt {a+b x+c x^2} \left (-128 a A c^2 e^3+220 a b B c e^3-384 a B c^2 d e^2-72 a B c^2 e^3 x+120 A b^2 c e^3-432 A b c^2 d e^2-80 A b c^2 e^3 x+576 A c^3 d^2 e+288 A c^3 d e^2 x+64 A c^3 e^3 x^2-105 b^3 B e^3+360 b^2 B c d e^2+70 b^2 B c e^3 x-432 b B c^2 d^2 e-240 b B c^2 d e^2 x-56 b B c^2 e^3 x^2+192 B c^3 d^3+288 B c^3 d^2 e x+192 B c^3 d e^2 x^2+48 B c^3 e^3 x^3\right )}{192 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x + c*x^2]*(192*B*c^3*d^3 - 432*b*B*c^2*d^2*e + 576*A*c^3*d^2*e + 360*b^2*B*c*d*e^2 - 432*A*b*c^2*
d*e^2 - 384*a*B*c^2*d*e^2 - 105*b^3*B*e^3 + 120*A*b^2*c*e^3 + 220*a*b*B*c*e^3 - 128*a*A*c^2*e^3 + 288*B*c^3*d^
2*e*x - 240*b*B*c^2*d*e^2*x + 288*A*c^3*d*e^2*x + 70*b^2*B*c*e^3*x - 80*A*b*c^2*e^3*x - 72*a*B*c^2*e^3*x + 192
*B*c^3*d*e^2*x^2 - 56*b*B*c^2*e^3*x^2 + 64*A*c^3*e^3*x^2 + 48*B*c^3*e^3*x^3))/(192*c^4) + ((64*b*B*c^3*d^3 - 1
28*A*c^4*d^3 - 144*b^2*B*c^2*d^2*e + 192*A*b*c^3*d^2*e + 192*a*B*c^3*d^2*e + 120*b^3*B*c*d*e^2 - 144*A*b^2*c^2
*d*e^2 - 288*a*b*B*c^2*d*e^2 + 192*a*A*c^3*d*e^2 - 35*b^4*B*e^3 + 40*A*b^3*c*e^3 + 120*a*b^2*B*c*e^3 - 96*a*A*
b*c^2*e^3 - 48*a^2*B*c^2*e^3)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + b*x + c*x^2]])/(128*c^(9/2))

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fricas [A]  time = 0.66, size = 811, normalized size = 1.99 \begin {gather*} \left [-\frac {3 \, {\left (64 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \, {\left (3 \, B b^{2} c^{2} - 4 \, {\left (B a + A b\right )} c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c + 8 \, A a c^{3} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c^{2}\right )} d e^{2} - {\left (35 \, B b^{4} + 48 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} - 40 \, {\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} e^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \, {\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 24 \, {\left (15 \, B b^{2} c^{2} - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{3}\right )} d e^{2} - {\left (105 \, B b^{3} c + 128 \, A a c^{3} - 20 \, {\left (11 \, B a b + 6 \, A b^{2}\right )} c^{2}\right )} e^{3} + 8 \, {\left (24 \, B c^{4} d e^{2} - {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (144 \, B c^{4} d^{2} e - 24 \, {\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + {\left (35 \, B b^{2} c^{2} - 4 \, {\left (9 \, B a + 10 \, A b\right )} c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{5}}, \frac {3 \, {\left (64 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \, {\left (3 \, B b^{2} c^{2} - 4 \, {\left (B a + A b\right )} c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c + 8 \, A a c^{3} - 6 \, {\left (2 \, B a b + A b^{2}\right )} c^{2}\right )} d e^{2} - {\left (35 \, B b^{4} + 48 \, {\left (B a^{2} + 2 \, A a b\right )} c^{2} - 40 \, {\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \, {\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 24 \, {\left (15 \, B b^{2} c^{2} - 2 \, {\left (8 \, B a + 9 \, A b\right )} c^{3}\right )} d e^{2} - {\left (105 \, B b^{3} c + 128 \, A a c^{3} - 20 \, {\left (11 \, B a b + 6 \, A b^{2}\right )} c^{2}\right )} e^{3} + 8 \, {\left (24 \, B c^{4} d e^{2} - {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (144 \, B c^{4} d^{2} e - 24 \, {\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + {\left (35 \, B b^{2} c^{2} - 4 \, {\left (9 \, B a + 10 \, A b\right )} c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, c^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/768*(3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*(B*a + A*b)*c^3)*d^2*e + 24*(5*B*b^3*c + 8*A*a*c^
3 - 6*(2*B*a*b + A*b^2)*c^2)*d*e^2 - (35*B*b^4 + 48*(B*a^2 + 2*A*a*b)*c^2 - 40*(3*B*a*b^2 + A*b^3)*c)*e^3)*sqr
t(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*(48*B*c^4*e^3*x
^3 + 192*B*c^4*d^3 - 144*(3*B*b*c^3 - 4*A*c^4)*d^2*e + 24*(15*B*b^2*c^2 - 2*(8*B*a + 9*A*b)*c^3)*d*e^2 - (105*
B*b^3*c + 128*A*a*c^3 - 20*(11*B*a*b + 6*A*b^2)*c^2)*e^3 + 8*(24*B*c^4*d*e^2 - (7*B*b*c^3 - 8*A*c^4)*e^3)*x^2
+ 2*(144*B*c^4*d^2*e - 24*(5*B*b*c^3 - 6*A*c^4)*d*e^2 + (35*B*b^2*c^2 - 4*(9*B*a + 10*A*b)*c^3)*e^3)*x)*sqrt(c
*x^2 + b*x + a))/c^5, 1/384*(3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*(B*a + A*b)*c^3)*d^2*e + 24*(
5*B*b^3*c + 8*A*a*c^3 - 6*(2*B*a*b + A*b^2)*c^2)*d*e^2 - (35*B*b^4 + 48*(B*a^2 + 2*A*a*b)*c^2 - 40*(3*B*a*b^2
+ A*b^3)*c)*e^3)*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*(
48*B*c^4*e^3*x^3 + 192*B*c^4*d^3 - 144*(3*B*b*c^3 - 4*A*c^4)*d^2*e + 24*(15*B*b^2*c^2 - 2*(8*B*a + 9*A*b)*c^3)
*d*e^2 - (105*B*b^3*c + 128*A*a*c^3 - 20*(11*B*a*b + 6*A*b^2)*c^2)*e^3 + 8*(24*B*c^4*d*e^2 - (7*B*b*c^3 - 8*A*
c^4)*e^3)*x^2 + 2*(144*B*c^4*d^2*e - 24*(5*B*b*c^3 - 6*A*c^4)*d*e^2 + (35*B*b^2*c^2 - 4*(9*B*a + 10*A*b)*c^3)*
e^3)*x)*sqrt(c*x^2 + b*x + a))/c^5]

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giac [A]  time = 0.30, size = 412, normalized size = 1.01 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (\frac {6 \, B x e^{3}}{c} + \frac {24 \, B c^{3} d e^{2} - 7 \, B b c^{2} e^{3} + 8 \, A c^{3} e^{3}}{c^{4}}\right )} x + \frac {144 \, B c^{3} d^{2} e - 120 \, B b c^{2} d e^{2} + 144 \, A c^{3} d e^{2} + 35 \, B b^{2} c e^{3} - 36 \, B a c^{2} e^{3} - 40 \, A b c^{2} e^{3}}{c^{4}}\right )} x + \frac {192 \, B c^{3} d^{3} - 432 \, B b c^{2} d^{2} e + 576 \, A c^{3} d^{2} e + 360 \, B b^{2} c d e^{2} - 384 \, B a c^{2} d e^{2} - 432 \, A b c^{2} d e^{2} - 105 \, B b^{3} e^{3} + 220 \, B a b c e^{3} + 120 \, A b^{2} c e^{3} - 128 \, A a c^{2} e^{3}}{c^{4}}\right )} + \frac {{\left (64 \, B b c^{3} d^{3} - 128 \, A c^{4} d^{3} - 144 \, B b^{2} c^{2} d^{2} e + 192 \, B a c^{3} d^{2} e + 192 \, A b c^{3} d^{2} e + 120 \, B b^{3} c d e^{2} - 288 \, B a b c^{2} d e^{2} - 144 \, A b^{2} c^{2} d e^{2} + 192 \, A a c^{3} d e^{2} - 35 \, B b^{4} e^{3} + 120 \, B a b^{2} c e^{3} + 40 \, A b^{3} c e^{3} - 48 \, B a^{2} c^{2} e^{3} - 96 \, A a b c^{2} e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*B*x*e^3/c + (24*B*c^3*d*e^2 - 7*B*b*c^2*e^3 + 8*A*c^3*e^3)/c^4)*x + (144*
B*c^3*d^2*e - 120*B*b*c^2*d*e^2 + 144*A*c^3*d*e^2 + 35*B*b^2*c*e^3 - 36*B*a*c^2*e^3 - 40*A*b*c^2*e^3)/c^4)*x +
(192*B*c^3*d^3 - 432*B*b*c^2*d^2*e + 576*A*c^3*d^2*e + 360*B*b^2*c*d*e^2 - 384*B*a*c^2*d*e^2 - 432*A*b*c^2*d*
e^2 - 105*B*b^3*e^3 + 220*B*a*b*c*e^3 + 120*A*b^2*c*e^3 - 128*A*a*c^2*e^3)/c^4) + 1/128*(64*B*b*c^3*d^3 - 128*
A*c^4*d^3 - 144*B*b^2*c^2*d^2*e + 192*B*a*c^3*d^2*e + 192*A*b*c^3*d^2*e + 120*B*b^3*c*d*e^2 - 288*B*a*b*c^2*d*
e^2 - 144*A*b^2*c^2*d*e^2 + 192*A*a*c^3*d*e^2 - 35*B*b^4*e^3 + 120*B*a*b^2*c*e^3 + 40*A*b^3*c*e^3 - 48*B*a^2*c
^2*e^3 - 96*A*a*b*c^2*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(9/2)

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maple [B]  time = 0.06, size = 981, normalized size = 2.41 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x +a}\, B \,e^{3} x^{3}}{4 c}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,e^{3} x^{2}}{3 c}-\frac {7 \sqrt {c \,x^{2}+b x +a}\, B b \,e^{3} x^{2}}{24 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B d \,e^{2} x^{2}}{c}+\frac {3 A a b \,e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}-\frac {3 A a d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}-\frac {5 A \,b^{3} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {9 A \,b^{2} d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {3 A b \,d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}+\frac {A \,d^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+\frac {3 B \,a^{2} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {15 B a \,b^{2} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {9 B a b d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4 c^{\frac {5}{2}}}-\frac {3 B a \,d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}+\frac {35 B \,b^{4} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{128 c^{\frac {9}{2}}}-\frac {15 B \,b^{3} d \,e^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 c^{\frac {7}{2}}}+\frac {9 B \,b^{2} d^{2} e \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {5}{2}}}-\frac {B b \,d^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, A b \,e^{3} x}{12 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A d \,e^{2} x}{2 c}-\frac {3 \sqrt {c \,x^{2}+b x +a}\, B a \,e^{3} x}{8 c^{2}}+\frac {35 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} e^{3} x}{96 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x +a}\, B b d \,e^{2} x}{4 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,d^{2} e x}{2 c}-\frac {2 \sqrt {c \,x^{2}+b x +a}\, A a \,e^{3}}{3 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2} e^{3}}{8 c^{3}}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, A b d \,e^{2}}{4 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,d^{2} e}{c}+\frac {55 \sqrt {c \,x^{2}+b x +a}\, B a b \,e^{3}}{48 c^{3}}-\frac {2 \sqrt {c \,x^{2}+b x +a}\, B a d \,e^{2}}{c^{2}}-\frac {35 \sqrt {c \,x^{2}+b x +a}\, B \,b^{3} e^{3}}{64 c^{4}}+\frac {15 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2} d \,e^{2}}{8 c^{3}}-\frac {9 \sqrt {c \,x^{2}+b x +a}\, B b \,d^{2} e}{4 c^{2}}+\frac {\sqrt {c \,x^{2}+b x +a}\, B \,d^{3}}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(1/2),x)

[Out]

-5/4*b/c^2*x*(c*x^2+b*x+a)^(1/2)*B*d*e^2+9/4*b/c^(5/2)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d*e^2+x
^2/c*(c*x^2+b*x+a)^(1/2)*B*d*e^2-3/8*B*e^3*a/c^2*x*(c*x^2+b*x+a)^(1/2)-7/24*B*e^3*b/c^2*x^2*(c*x^2+b*x+a)^(1/2
)+35/96*B*e^3*b^2/c^3*x*(c*x^2+b*x+a)^(1/2)-15/16*B*e^3*b^2/c^(7/2)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/
2))+55/48*B*e^3*b/c^3*a*(c*x^2+b*x+a)^(1/2)+3/4*b/c^(5/2)*a*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*A*e^3-
2*a/c^2*(c*x^2+b*x+a)^(1/2)*B*d*e^2+3/2*x/c*(c*x^2+b*x+a)^(1/2)*A*d*e^2+3/2*x/c*(c*x^2+b*x+a)^(1/2)*B*d^2*e-5/
12*b/c^2*x*(c*x^2+b*x+a)^(1/2)*A*e^3+15/8*b^2/c^3*(c*x^2+b*x+a)^(1/2)*B*d*e^2-15/16*b^3/c^(7/2)*ln((c*x+1/2*b)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d*e^2-3/2*a/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*A*d*e^2+9/8*b^
2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d^2*e-3/2*b/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*A*d^2*e-3/2*a/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d^2*e-9/4*b/c^2*(c*x^2+b*x+a)^(1
/2)*A*d*e^2-9/4*b/c^2*(c*x^2+b*x+a)^(1/2)*B*d^2*e+9/8*b^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*
A*d*e^2+A*d^3*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+1/c*(c*x^2+b*x+a)^(1/2)*B*d^3+1/3*x^2/c*(c*x
^2+b*x+a)^(1/2)*A*e^3+5/8*b^2/c^3*(c*x^2+b*x+a)^(1/2)*A*e^3-35/64*B*e^3*b^3/c^4*(c*x^2+b*x+a)^(1/2)+35/128*B*e
^3*b^4/c^(9/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/8*B*e^3*a^2/c^(5/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2
+b*x+a)^(1/2))-1/2*b/c^(3/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*B*d^3+1/4*B*e^3*x^3/c*(c*x^2+b*x+a)^(
1/2)-5/16*b^3/c^(7/2)*ln((c*x+1/2*b)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*A*e^3-2/3*a/c^2*(c*x^2+b*x+a)^(1/2)*A*e^3+3/
c*(c*x^2+b*x+a)^(1/2)*A*d^2*e

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^3/(c*x^2+b*x+a)^(1/2),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 details)Is 4*a*c-b^2 zero or nonzero?

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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 )}^3}{\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(1/2),x)

[Out]

int(((A + B*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(1/2), 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 )^{3}}{\sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**3/(c*x**2+b*x+a)**(1/2),x)

[Out]

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

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