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

Optimal. Leaf size=314 \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{4 \sqrt {c} e^5}+\frac {3 \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 \sqrt {d} e^5 \sqrt {c d-b e}}-\frac {3 \sqrt {b x+c x^2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{4 e^4 (d+e x)}+\frac {\left (b x+c x^2\right )^{3/2} (-A e+2 B d+B e x)}{2 e^2 (d+e x)^2} \]

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Rubi [A]  time = 0.43, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {812, 843, 620, 206, 724} \begin {gather*} -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{4 \sqrt {c} e^5}+\frac {3 \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 \sqrt {d} e^5 \sqrt {c d-b e}}+\frac {\left (b x+c x^2\right )^{3/2} (-A e+2 B d+B e x)}{2 e^2 (d+e x)^2}-\frac {3 \sqrt {b x+c x^2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{4 e^4 (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(4*B*d*(2*c*d - b*e) - A*e*(4*c*d - b*e) + e*(4*B*c*d - b*B*e - 2*A*c*e)*x)*Sqrt[b*x + c*x^2])/(4*e^4*(d +
e*x)) + ((2*B*d - A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(2*e^2*(d + e*x)^2) - (3*(4*A*c*e*(2*c*d - b*e) - B*(16*c
^2*d^2 - 12*b*c*d*e + b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*Sqrt[c]*e^5) + (3*(A*e*(8*c^2*d^2 -
8*b*c*d*e + b^2*e^2) - B*d*(16*c^2*d^2 - 20*b*c*d*e + 5*b^2*e^2))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*
Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*Sqrt[d]*e^5*Sqrt[c*d - b*e])

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 620

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

Rule 724

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

Rule 812

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

Rule 843

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

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx &=\frac {(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac {3 \int \frac {(2 b (2 B d-A e)+2 (4 B c d-b B e-2 A c e) x) \sqrt {b x+c x^2}}{(d+e x)^2} \, dx}{8 e^2}\\ &=-\frac {3 (4 B d (2 c d-b e)-A e (4 c d-b e)+e (4 B c d-b B e-2 A c e) x) \sqrt {b x+c x^2}}{4 e^4 (d+e x)}+\frac {(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}+\frac {3 \int \frac {2 b (4 B d (2 c d-b e)-A e (4 c d-b e))-2 \left (4 A c e (2 c d-b e)-B \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\right ) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{16 e^4}\\ &=-\frac {3 (4 B d (2 c d-b e)-A e (4 c d-b e)+e (4 B c d-b B e-2 A c e) x) \sqrt {b x+c x^2}}{4 e^4 (d+e x)}+\frac {(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac {\left (3 \left (4 A c e (2 c d-b e)-B \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\right )\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{8 e^5}+\frac {\left (3 \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 e^5}\\ &=-\frac {3 (4 B d (2 c d-b e)-A e (4 c d-b e)+e (4 B c d-b B e-2 A c e) x) \sqrt {b x+c x^2}}{4 e^4 (d+e x)}+\frac {(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac {\left (3 \left (4 A c e (2 c d-b e)-B \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 e^5}-\frac {\left (3 \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac {-b d-(2 c d-b e) x}{\sqrt {b x+c x^2}}\right )}{4 e^5}\\ &=-\frac {3 (4 B d (2 c d-b e)-A e (4 c d-b e)+e (4 B c d-b B e-2 A c e) x) \sqrt {b x+c x^2}}{4 e^4 (d+e x)}+\frac {(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac {3 \left (4 A c e (2 c d-b e)-B \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {c} e^5}+\frac {3 \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 \sqrt {d} e^5 \sqrt {c d-b e}}\\ \end {align*}

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Mathematica [B]  time = 6.12, size = 1358, normalized size = 4.32

result too large to display

Antiderivative was successfully verified.

[In]

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

[Out]

((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(3/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2) + ((x*(b + c*x))^(3/2)*(((-3
*c*d*(B*d - A*e) + (e*(5*b*B*d - 4*A*c*d - A*b*e))/2)*x^(5/2)*(b + c*x)^(5/2))/(d*(-(c*d) + b*e)*(d + e*x)) +
(((8*A*c^2*d^2 + 4*b*c*d*(5*B*d - 4*A*e) - 3*b^2*e*(5*B*d - A*e))*((2*b*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^2*
((3/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 + (3*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt
[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(32*c^2*x^2*(1 + (c*x)/b)^2)))/(3*e) - (d*((2*b*Sqrt[x]*Sqrt[b +
c*x]*(1 + (c*x)/b)^2*((3/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/4 + (3*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sq
rt[b]])/(8*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(5/2))))/e - (d*((2*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)*(1/(2*(1 +
(c*x)/b)) + (Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(2*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2))))/e - ((c*d -
b*e)*((2*Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(e*Sqrt[b + c*x]) - (2*Sqrt[c*
d - b*e]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e)))/e))/e))/e))/4 + 2*c*(B*d*(6
*c*d - 5*b*e) - A*e*(2*c*d - b*e))*((2*b*x^(5/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^2*((5*(1/(2*(1 + (c*x)/b)^2) + (1
+ (c*x)/b)^(-1)))/8 - (15*b^3*((2*c*x)/b - (4*c^2*x^2)/(3*b^2) - (2*Sqrt[c]*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])
/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(256*c^3*x^3*(1 + (c*x)/b)^2)))/(5*e) - (d*((2*b*x^(3/2)*Sqrt[b + c*x
]*(1 + (c*x)/b)^2*((3/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 + (3*b^2*((2*c*x)/b - (2*Sqrt[c]*Sqrt[x]*Arc
Sinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(32*c^2*x^2*(1 + (c*x)/b)^2)))/(3*e) - (d*((2*b
*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)^2*((3/(2*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/4 + (3*Sqrt[b]*ArcSinh[(S
qrt[c]*Sqrt[x])/Sqrt[b]])/(8*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(5/2))))/e - (d*((2*c*Sqrt[x]*Sqrt[b + c*x]*(1 + (c
*x)/b)*(1/(2*(1 + (c*x)/b)) + (Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(2*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3
/2))))/e - ((c*d - b*e)*((2*Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(e*Sqrt[b +
c*x]) - (2*Sqrt[c*d - b*e]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e)))/e))/e))/e
))/e))/(d*(-(c*d) + b*e))))/(2*d*(-(c*d) + b*e)*x^(3/2)*(b + c*x)^(3/2))

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IntegrateAlgebraic [A]  time = 5.22, size = 348, normalized size = 1.11 \begin {gather*} -\frac {3 \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (4 A b c e^2-8 A c^2 d e+b^2 B e^2-12 b B c d e+16 B c^2 d^2\right )}{8 \sqrt {c} e^5}-\frac {3 \left (-A b^2 e^3+8 A b c d e^2-8 A c^2 d^2 e+5 b^2 B d e^2-20 b B c d^2 e+16 B c^2 d^3\right ) \tanh ^{-1}\left (\frac {-e \sqrt {b x+c x^2}+\sqrt {c} d+\sqrt {c} e x}{\sqrt {d} \sqrt {c d-b e}}\right )}{4 \sqrt {d} e^5 \sqrt {c d-b e}}+\frac {\sqrt {b x+c x^2} \left (-3 A b d e^2-5 A b e^3 x+12 A c d^2 e+18 A c d e^2 x+4 A c e^3 x^2+12 b B d^2 e+19 b B d e^2 x+5 b B e^3 x^2-24 B c d^3-36 B c d^2 e x-8 B c d e^2 x^2+2 B c e^3 x^3\right )}{4 e^4 (d+e x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b*x + c*x^2]*(-24*B*c*d^3 + 12*b*B*d^2*e + 12*A*c*d^2*e - 3*A*b*d*e^2 - 36*B*c*d^2*e*x + 19*b*B*d*e^2*x
+ 18*A*c*d*e^2*x - 5*A*b*e^3*x - 8*B*c*d*e^2*x^2 + 5*b*B*e^3*x^2 + 4*A*c*e^3*x^2 + 2*B*c*e^3*x^3))/(4*e^4*(d +
e*x)^2) - (3*(16*B*c^2*d^3 - 20*b*B*c*d^2*e - 8*A*c^2*d^2*e + 5*b^2*B*d*e^2 + 8*A*b*c*d*e^2 - A*b^2*e^3)*ArcT
anh[(Sqrt[c]*d + Sqrt[c]*e*x - e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(4*Sqrt[d]*e^5*Sqrt[c*d - b*e]
) - (3*(16*B*c^2*d^2 - 12*b*B*c*d*e - 8*A*c^2*d*e + b^2*B*e^2 + 4*A*b*c*e^2)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*
x + c*x^2]])/(8*Sqrt[c]*e^5)

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fricas [B]  time = 1.85, size = 3342, normalized size = 10.64

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(3*(16*B*c^3*d^6 - 4*(7*B*b*c^2 + 2*A*c^3)*d^5*e + (13*B*b^2*c + 12*A*b*c^2)*d^4*e^2 - (B*b^3 + 4*A*b^2*c
)*d^3*e^3 + (16*B*c^3*d^4*e^2 - 4*(7*B*b*c^2 + 2*A*c^3)*d^3*e^3 + (13*B*b^2*c + 12*A*b*c^2)*d^2*e^4 - (B*b^3 +
4*A*b^2*c)*d*e^5)*x^2 + 2*(16*B*c^3*d^5*e - 4*(7*B*b*c^2 + 2*A*c^3)*d^4*e^2 + (13*B*b^2*c + 12*A*b*c^2)*d^3*e
^3 - (B*b^3 + 4*A*b^2*c)*d^2*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 3*(16*B*c^3*d^5 -
A*b^2*c*d^2*e^3 - 4*(5*B*b*c^2 + 2*A*c^3)*d^4*e + (5*B*b^2*c + 8*A*b*c^2)*d^3*e^2 + (16*B*c^3*d^3*e^2 - A*b^2*
c*e^5 - 4*(5*B*b*c^2 + 2*A*c^3)*d^2*e^3 + (5*B*b^2*c + 8*A*b*c^2)*d*e^4)*x^2 + 2*(16*B*c^3*d^4*e - A*b^2*c*d*e
^4 - 4*(5*B*b*c^2 + 2*A*c^3)*d^3*e^2 + (5*B*b^2*c + 8*A*b*c^2)*d^2*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c
*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) - 2*(24*B*c^3*d^5*e - 3*A*b^2*c*d^2*e^4 - 12
*(3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(4*B*b^2*c + 5*A*b*c^2)*d^3*e^3 - 2*(B*c^3*d^2*e^4 - B*b*c^2*d*e^5)*x^3 + (8*
B*c^3*d^3*e^3 - (13*B*b*c^2 + 4*A*c^3)*d^2*e^4 + (5*B*b^2*c + 4*A*b*c^2)*d*e^5)*x^2 + (36*B*c^3*d^4*e^2 - 5*A*
b^2*c*d*e^5 - (55*B*b*c^2 + 18*A*c^3)*d^3*e^3 + (19*B*b^2*c + 23*A*b*c^2)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*
d^4*e^5 - b*c*d^3*e^6 + (c^2*d^2*e^7 - b*c*d*e^8)*x^2 + 2*(c^2*d^3*e^6 - b*c*d^2*e^7)*x), -1/8*(6*(16*B*c^3*d^
5 - A*b^2*c*d^2*e^3 - 4*(5*B*b*c^2 + 2*A*c^3)*d^4*e + (5*B*b^2*c + 8*A*b*c^2)*d^3*e^2 + (16*B*c^3*d^3*e^2 - A*
b^2*c*e^5 - 4*(5*B*b*c^2 + 2*A*c^3)*d^2*e^3 + (5*B*b^2*c + 8*A*b*c^2)*d*e^4)*x^2 + 2*(16*B*c^3*d^4*e - A*b^2*c
*d*e^4 - 4*(5*B*b*c^2 + 2*A*c^3)*d^3*e^2 + (5*B*b^2*c + 8*A*b*c^2)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sq
rt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - 3*(16*B*c^3*d^6 - 4*(7*B*b*c^2 + 2*A*c^3)*d^5*e + (13*
B*b^2*c + 12*A*b*c^2)*d^4*e^2 - (B*b^3 + 4*A*b^2*c)*d^3*e^3 + (16*B*c^3*d^4*e^2 - 4*(7*B*b*c^2 + 2*A*c^3)*d^3*
e^3 + (13*B*b^2*c + 12*A*b*c^2)*d^2*e^4 - (B*b^3 + 4*A*b^2*c)*d*e^5)*x^2 + 2*(16*B*c^3*d^5*e - 4*(7*B*b*c^2 +
2*A*c^3)*d^4*e^2 + (13*B*b^2*c + 12*A*b*c^2)*d^3*e^3 - (B*b^3 + 4*A*b^2*c)*d^2*e^4)*x)*sqrt(c)*log(2*c*x + b +
2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(24*B*c^3*d^5*e - 3*A*b^2*c*d^2*e^4 - 12*(3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(4*
B*b^2*c + 5*A*b*c^2)*d^3*e^3 - 2*(B*c^3*d^2*e^4 - B*b*c^2*d*e^5)*x^3 + (8*B*c^3*d^3*e^3 - (13*B*b*c^2 + 4*A*c^
3)*d^2*e^4 + (5*B*b^2*c + 4*A*b*c^2)*d*e^5)*x^2 + (36*B*c^3*d^4*e^2 - 5*A*b^2*c*d*e^5 - (55*B*b*c^2 + 18*A*c^3
)*d^3*e^3 + (19*B*b^2*c + 23*A*b*c^2)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^4*e^5 - b*c*d^3*e^6 + (c^2*d^2*e^7
- b*c*d*e^8)*x^2 + 2*(c^2*d^3*e^6 - b*c*d^2*e^7)*x), -1/8*(6*(16*B*c^3*d^6 - 4*(7*B*b*c^2 + 2*A*c^3)*d^5*e +
(13*B*b^2*c + 12*A*b*c^2)*d^4*e^2 - (B*b^3 + 4*A*b^2*c)*d^3*e^3 + (16*B*c^3*d^4*e^2 - 4*(7*B*b*c^2 + 2*A*c^3)*
d^3*e^3 + (13*B*b^2*c + 12*A*b*c^2)*d^2*e^4 - (B*b^3 + 4*A*b^2*c)*d*e^5)*x^2 + 2*(16*B*c^3*d^5*e - 4*(7*B*b*c^
2 + 2*A*c^3)*d^4*e^2 + (13*B*b^2*c + 12*A*b*c^2)*d^3*e^3 - (B*b^3 + 4*A*b^2*c)*d^2*e^4)*x)*sqrt(-c)*arctan(sqr
t(c*x^2 + b*x)*sqrt(-c)/(c*x)) - 3*(16*B*c^3*d^5 - A*b^2*c*d^2*e^3 - 4*(5*B*b*c^2 + 2*A*c^3)*d^4*e + (5*B*b^2*
c + 8*A*b*c^2)*d^3*e^2 + (16*B*c^3*d^3*e^2 - A*b^2*c*e^5 - 4*(5*B*b*c^2 + 2*A*c^3)*d^2*e^3 + (5*B*b^2*c + 8*A*
b*c^2)*d*e^4)*x^2 + 2*(16*B*c^3*d^4*e - A*b^2*c*d*e^4 - 4*(5*B*b*c^2 + 2*A*c^3)*d^3*e^2 + (5*B*b^2*c + 8*A*b*c
^2)*d^2*e^3)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x
+ d)) + 2*(24*B*c^3*d^5*e - 3*A*b^2*c*d^2*e^4 - 12*(3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(4*B*b^2*c + 5*A*b*c^2)*d^
3*e^3 - 2*(B*c^3*d^2*e^4 - B*b*c^2*d*e^5)*x^3 + (8*B*c^3*d^3*e^3 - (13*B*b*c^2 + 4*A*c^3)*d^2*e^4 + (5*B*b^2*c
+ 4*A*b*c^2)*d*e^5)*x^2 + (36*B*c^3*d^4*e^2 - 5*A*b^2*c*d*e^5 - (55*B*b*c^2 + 18*A*c^3)*d^3*e^3 + (19*B*b^2*c
+ 23*A*b*c^2)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^4*e^5 - b*c*d^3*e^6 + (c^2*d^2*e^7 - b*c*d*e^8)*x^2 + 2*(
c^2*d^3*e^6 - b*c*d^2*e^7)*x), -1/4*(3*(16*B*c^3*d^5 - A*b^2*c*d^2*e^3 - 4*(5*B*b*c^2 + 2*A*c^3)*d^4*e + (5*B*
b^2*c + 8*A*b*c^2)*d^3*e^2 + (16*B*c^3*d^3*e^2 - A*b^2*c*e^5 - 4*(5*B*b*c^2 + 2*A*c^3)*d^2*e^3 + (5*B*b^2*c +
8*A*b*c^2)*d*e^4)*x^2 + 2*(16*B*c^3*d^4*e - A*b^2*c*d*e^4 - 4*(5*B*b*c^2 + 2*A*c^3)*d^3*e^2 + (5*B*b^2*c + 8*A
*b*c^2)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + 3*(
16*B*c^3*d^6 - 4*(7*B*b*c^2 + 2*A*c^3)*d^5*e + (13*B*b^2*c + 12*A*b*c^2)*d^4*e^2 - (B*b^3 + 4*A*b^2*c)*d^3*e^3
+ (16*B*c^3*d^4*e^2 - 4*(7*B*b*c^2 + 2*A*c^3)*d^3*e^3 + (13*B*b^2*c + 12*A*b*c^2)*d^2*e^4 - (B*b^3 + 4*A*b^2*
c)*d*e^5)*x^2 + 2*(16*B*c^3*d^5*e - 4*(7*B*b*c^2 + 2*A*c^3)*d^4*e^2 + (13*B*b^2*c + 12*A*b*c^2)*d^3*e^3 - (B*b
^3 + 4*A*b^2*c)*d^2*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (24*B*c^3*d^5*e - 3*A*b^2*c*d^
2*e^4 - 12*(3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(4*B*b^2*c + 5*A*b*c^2)*d^3*e^3 - 2*(B*c^3*d^2*e^4 - B*b*c^2*d*e^5)
*x^3 + (8*B*c^3*d^3*e^3 - (13*B*b*c^2 + 4*A*c^3)*d^2*e^4 + (5*B*b^2*c + 4*A*b*c^2)*d*e^5)*x^2 + (36*B*c^3*d^4*
e^2 - 5*A*b^2*c*d*e^5 - (55*B*b*c^2 + 18*A*c^3)*d^3*e^3 + (19*B*b^2*c + 23*A*b*c^2)*d^2*e^4)*x)*sqrt(c*x^2 + b
*x))/(c^2*d^4*e^5 - b*c*d^3*e^6 + (c^2*d^2*e^7 - b*c*d*e^8)*x^2 + 2*(c^2*d^3*e^6 - b*c*d^2*e^7)*x)]

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giac [B]  time = 0.52, size = 936, normalized size = 2.98 \begin {gather*} -\frac {3 \, {\left (16 \, B c^{2} d^{3} - 20 \, B b c d^{2} e - 8 \, A c^{2} d^{2} e + 5 \, B b^{2} d e^{2} + 8 \, A b c d e^{2} - A b^{2} e^{3}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right ) e^{\left (-5\right )}}{4 \, \sqrt {-c d^{2} + b d e}} - \frac {3 \, {\left (16 \, B c^{2} d^{2} - 12 \, B b c d e - 8 \, A c^{2} d e + B b^{2} e^{2} + 4 \, A b c e^{2}\right )} e^{\left (-5\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{8 \, \sqrt {c}} + \frac {1}{4} \, {\left (2 \, B c x e^{\left (-3\right )} - \frac {{\left (12 \, B c^{2} d e^{8} - 5 \, B b c e^{9} - 4 \, A c^{2} e^{9}\right )} e^{\left (-12\right )}}{c}\right )} \sqrt {c x^{2} + b x} - \frac {{\left (32 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B c^{\frac {5}{2}} d^{3} e + 56 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B c^{3} d^{4} - 44 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b c^{2} d^{3} e - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c^{3} d^{3} e + 56 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b c^{\frac {5}{2}} d^{4} - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b c^{\frac {3}{2}} d^{2} e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{\frac {5}{2}} d^{2} e^{2} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{2} c^{\frac {3}{2}} d^{3} e - 40 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b c^{\frac {5}{2}} d^{3} e + 14 \, B b^{2} c^{2} d^{4} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} c d^{2} e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c^{2} d^{2} e^{2} - 9 \, B b^{3} c d^{3} e - 10 \, A b^{2} c^{2} d^{3} e + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} \sqrt {c} d e^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c^{\frac {3}{2}} d e^{3} + 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{3} \sqrt {c} d^{2} e^{2} + 28 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} c^{\frac {3}{2}} d^{2} e^{2} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} c d e^{3} + 5 \, A b^{3} c d^{2} e^{2} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} \sqrt {c} e^{4} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} \sqrt {c} d e^{3}\right )} e^{\left (-5\right )}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2} \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(16*B*c^2*d^3 - 20*B*b*c*d^2*e - 8*A*c^2*d^2*e + 5*B*b^2*d*e^2 + 8*A*b*c*d*e^2 - A*b^2*e^3)*arctan(-((sqr
t(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))*e^(-5)/sqrt(-c*d^2 + b*d*e) - 3/8*(16*B*c^2*d
^2 - 12*B*b*c*d*e - 8*A*c^2*d*e + B*b^2*e^2 + 4*A*b*c*e^2)*e^(-5)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sq
rt(c) + b))/sqrt(c) + 1/4*(2*B*c*x*e^(-3) - (12*B*c^2*d*e^8 - 5*B*b*c*e^9 - 4*A*c^2*e^9)*e^(-12)/c)*sqrt(c*x^2
+ b*x) - 1/4*(32*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^(5/2)*d^3*e + 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*c
^3*d^4 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^2*d^3*e - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*c^3*d^3*e
+ 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^(5/2)*d^4 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c^(3/2)*d^2*e
^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(5/2)*d^2*e^2 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^2*c^(3/2)
*d^3*e - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*c^(5/2)*d^3*e + 14*B*b^2*c^2*d^4 + 3*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^2*B*b^2*c*d^2*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^2*d^2*e^2 - 9*B*b^3*c*d^3*e - 10*A*b^2*c
^2*d^3*e + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*sqrt(c)*d*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*
c^(3/2)*d*e^3 + 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*sqrt(c)*d^2*e^2 + 28*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A
*b^2*c^(3/2)*d^2*e^2 + (sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c*d*e^3 + 5*A*b^3*c*d^2*e^2 - 5*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^3*A*b^2*sqrt(c)*e^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*sqrt(c)*d*e^3)*e^(-5)/(((sqrt(c
)*x - sqrt(c*x^2 + b*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2*sqrt(c))

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maple [B]  time = 0.06, size = 7365, normalized size = 23.46 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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)*(c*x^2+b*x)^(3/2)/(e*x+d)^3,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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d zero or nonzero?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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