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

Optimal. Leaf size=235 \[ -\frac {\left (A b^2 e^3+B d \left (3 b^2 e^2-12 b c d e+8 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{3/2} e^3 (c d-b e)^{3/2}}+\frac {\sqrt {b x+c x^2} \left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e x (B d (6 c d-5 b e)-A e (2 c d-b e))\right )}{4 d e^2 (d+e x)^2 (c d-b e)}+\frac {2 B \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^3} \]

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Rubi [A]  time = 0.32, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {810, 843, 620, 206, 724} \begin {gather*} -\frac {\left (A b^2 e^3+B d \left (3 b^2 e^2-12 b c d e+8 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 d^{3/2} e^3 (c d-b e)^{3/2}}+\frac {\sqrt {b x+c x^2} \left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e x (B d (6 c d-5 b e)-A e (2 c d-b e))\right )}{4 d e^2 (d+e x)^2 (c d-b e)}+\frac {2 B \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 810

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*((d*g - e*f*(m + 2))*(c*d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*
f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2
- b*d*e + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x
+ c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1)
- c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1
) - b*(d*g*(m - 2*p) + e*f*(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] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

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) \sqrt {b x+c x^2}}{(d+e x)^3} \, dx &=\frac {\left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e (B d (6 c d-5 b e)-A e (2 c d-b e)) x\right ) \sqrt {b x+c x^2}}{4 d e^2 (c d-b e) (d+e x)^2}-\frac {\int \frac {\frac {1}{2} b \left (A b e^2-B d (4 c d-3 b e)\right )-4 B c d (c d-b e) x}{(d+e x) \sqrt {b x+c x^2}} \, dx}{4 d e^2 (c d-b e)}\\ &=\frac {\left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e (B d (6 c d-5 b e)-A e (2 c d-b e)) x\right ) \sqrt {b x+c x^2}}{4 d e^2 (c d-b e) (d+e x)^2}+\frac {(B c) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{e^3}-\frac {\left (A b^2 e^3+B d \left (8 c^2 d^2-12 b c d e+3 b^2 e^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {b x+c x^2}} \, dx}{8 d e^3 (c d-b e)}\\ &=\frac {\left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e (B d (6 c d-5 b e)-A e (2 c d-b e)) x\right ) \sqrt {b x+c x^2}}{4 d e^2 (c d-b e) (d+e x)^2}+\frac {(2 B c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{e^3}+\frac {\left (A b^2 e^3+B d \left (8 c^2 d^2-12 b c d e+3 b^2 e^2\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 d e^3 (c d-b e)}\\ &=\frac {\left (d \left (A b e^2-B d (4 c d-3 b e)\right )-e (B d (6 c d-5 b e)-A e (2 c d-b e)) x\right ) \sqrt {b x+c x^2}}{4 d e^2 (c d-b e) (d+e x)^2}+\frac {2 B \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^3}-\frac {\left (A b^2 e^3+B d \left (8 c^2 d^2-12 b c d e+3 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 d^{3/2} e^3 (c d-b e)^{3/2}}\\ \end {align*}

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Mathematica [B]  time = 3.65, size = 690, normalized size = 2.94 \begin {gather*} \frac {x \left (\frac {(d+e x) \left ((d+e x) \left (b \sqrt {c} e \sqrt {x} \left (A \left (b^2 e^2+4 b c d e-8 c^2 d^2\right )+3 b^2 B d e\right ) \left (e (b+c x) \left (b \sqrt {c} \sqrt {x} \left (\frac {c x}{b}+1\right )^{3/2}+\sqrt {b} (b+c x) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )-2 b \sqrt {c} \sqrt {d} \left (\frac {c x}{b}+1\right )^{3/2} \left (\sqrt {b} \sqrt {c} \sqrt {d} \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )-\sqrt {b+c x} \sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )-(A e (2 c d-b e)+B d (2 c d-3 b e)) \left (e^2 (b+c x)^2 \left (b c x (b+2 c x) \sqrt {\frac {c x}{b}+1}-b^{5/2} \sqrt {c} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )-4 b c^{3/2} d \sqrt {x} \left (e (b+c x) \left (b \sqrt {c} \sqrt {x} \left (\frac {c x}{b}+1\right )^{3/2}+\sqrt {b} (b+c x) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )\right )-2 b \sqrt {c} \sqrt {d} \left (\frac {c x}{b}+1\right )^{3/2} \left (\sqrt {b} \sqrt {c} \sqrt {d} \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )-\sqrt {b+c x} \sqrt {c d-b e} \tanh ^{-1}\left (\frac {\sqrt {x} \sqrt {c d-b e}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )\right )\right )-2 b c e^3 x^2 (b+c x)^3 \sqrt {\frac {c x}{b}+1} (A e (b e-2 c d)+B d (3 b e-2 c d))\right )}{b c d e^3 \sqrt {\frac {c x}{b}+1} (c d-b e)}-4 x^2 (b+c x)^3 (B d-A e)\right )}{8 d (x (b+c x))^{3/2} (d+e x)^2 (b e-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(-4*(B*d - A*e)*x^2*(b + c*x)^3 + ((d + e*x)*(-2*b*c*e^3*(A*e*(-2*c*d + b*e) + B*d*(-2*c*d + 3*b*e))*x^2*(b
+ c*x)^3*Sqrt[1 + (c*x)/b] + (d + e*x)*(b*Sqrt[c]*e*(3*b^2*B*d*e + A*(-8*c^2*d^2 + 4*b*c*d*e + b^2*e^2))*Sqrt
[x]*(e*(b + c*x)*(b*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(3/2) + Sqrt[b]*(b + c*x)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]
) - 2*b*Sqrt[c]*Sqrt[d]*(1 + (c*x)/b)^(3/2)*(Sqrt[b]*Sqrt[c]*Sqrt[d]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x
])/Sqrt[b]] - Sqrt[c*d - b*e]*Sqrt[b + c*x]*ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])) - (B*
d*(2*c*d - 3*b*e) + A*e*(2*c*d - b*e))*(e^2*(b + c*x)^2*(b*c*x*(b + 2*c*x)*Sqrt[1 + (c*x)/b] - b^(5/2)*Sqrt[c]
*Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) - 4*b*c^(3/2)*d*Sqrt[x]*(e*(b + c*x)*(b*Sqrt[c]*Sqrt[x]*(1 + (c*x
)/b)^(3/2) + Sqrt[b]*(b + c*x)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]]) - 2*b*Sqrt[c]*Sqrt[d]*(1 + (c*x)/b)^(3/2)*(
Sqrt[b]*Sqrt[c]*Sqrt[d]*Sqrt[1 + (c*x)/b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]] - Sqrt[c*d - b*e]*Sqrt[b + c*x]*A
rcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))))))/(b*c*d*e^3*(c*d - b*e)*Sqrt[1 + (c*x)/b])))/(8
*d*(-(c*d) + b*e)*(x*(b + c*x))^(3/2)*(d + e*x)^2)

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IntegrateAlgebraic [A]  time = 2.71, size = 250, normalized size = 1.06 \begin {gather*} \frac {\left (-A b^2 e^3-3 b^2 B d e^2+12 b B c d^2 e-8 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 d^{3/2} e^3 (c d-b e)^{3/2}}-\frac {\sqrt {b x+c x^2} \left (-A b d e^2+A b e^3 x-2 A c d e^2 x-3 b B d^2 e-5 b B d e^2 x+4 B c d^3+6 B c d^2 e x\right )}{4 d e^2 (d+e x)^2 (c d-b e)}-\frac {B \sqrt {c} \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((4*B*c*d^3 - 3*b*B*d^2*e - A*b*d*e^2 + 6*B*c*d^2*e*x - 5*b*B*d*e^2*x - 2*A*c*d*e^2*x + A*b*e^3*x)*Sqrt[b
*x + c*x^2])/(d*e^2*(c*d - b*e)*(d + e*x)^2) + ((-8*B*c^2*d^3 + 12*b*B*c*d^2*e - 3*b^2*B*d*e^2 - A*b^2*e^3)*Ar
cTanh[(Sqrt[c]*d + Sqrt[c]*e*x - e*Sqrt[b*x + c*x^2])/(Sqrt[d]*Sqrt[c*d - b*e])])/(4*d^(3/2)*e^3*(c*d - b*e)^(
3/2)) - (B*Sqrt[c]*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/e^3

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fricas [B]  time = 1.95, size = 2234, normalized size = 9.51

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)^(1/2)/(e*x+d)^3,x, algorithm="fricas")

[Out]

[1/8*(8*(B*c^2*d^6 - 2*B*b*c*d^5*e + B*b^2*d^4*e^2 + (B*c^2*d^4*e^2 - 2*B*b*c*d^3*e^3 + B*b^2*d^2*e^4)*x^2 + 2
*(B*c^2*d^5*e - 2*B*b*c*d^4*e^2 + B*b^2*d^3*e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (8*
B*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3*e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d
*e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e - 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*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*(4*B*c^2*d^5*e - 7*
B*b*c*d^4*e^2 + A*b^2*d^2*e^4 + (3*B*b^2 - A*b*c)*d^3*e^3 + (6*B*c^2*d^4*e^2 - A*b^2*d*e^5 - (11*B*b*c + 2*A*c
^2)*d^3*e^3 + (5*B*b^2 + 3*A*b*c)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^6*e^3 - 2*b*c*d^5*e^4 + b^2*d^4*e^5 +
(c^2*d^4*e^5 - 2*b*c*d^3*e^6 + b^2*d^2*e^7)*x^2 + 2*(c^2*d^5*e^4 - 2*b*c*d^4*e^5 + b^2*d^3*e^6)*x), -1/4*((8*B
*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3*e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d*
e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e - 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*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)) - 4*(B*c^2*d^6 - 2*B*b*c*d^5*e + B*b^2*d
^4*e^2 + (B*c^2*d^4*e^2 - 2*B*b*c*d^3*e^3 + B*b^2*d^2*e^4)*x^2 + 2*(B*c^2*d^5*e - 2*B*b*c*d^4*e^2 + B*b^2*d^3*
e^3)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + (4*B*c^2*d^5*e - 7*B*b*c*d^4*e^2 + A*b^2*d^2*e^
4 + (3*B*b^2 - A*b*c)*d^3*e^3 + (6*B*c^2*d^4*e^2 - A*b^2*d*e^5 - (11*B*b*c + 2*A*c^2)*d^3*e^3 + (5*B*b^2 + 3*A
*b*c)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^6*e^3 - 2*b*c*d^5*e^4 + b^2*d^4*e^5 + (c^2*d^4*e^5 - 2*b*c*d^3*e^6
+ b^2*d^2*e^7)*x^2 + 2*(c^2*d^5*e^4 - 2*b*c*d^4*e^5 + b^2*d^3*e^6)*x), -1/8*(16*(B*c^2*d^6 - 2*B*b*c*d^5*e +
B*b^2*d^4*e^2 + (B*c^2*d^4*e^2 - 2*B*b*c*d^3*e^3 + B*b^2*d^2*e^4)*x^2 + 2*(B*c^2*d^5*e - 2*B*b*c*d^4*e^2 + B*b
^2*d^3*e^3)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (8*B*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3
*e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d*e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e
- 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*x)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqr
t(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(4*B*c^2*d^5*e - 7*B*b*c*d^4*e^2 + A*b^2*d^2*e^4 + (3*B*b^2
- A*b*c)*d^3*e^3 + (6*B*c^2*d^4*e^2 - A*b^2*d*e^5 - (11*B*b*c + 2*A*c^2)*d^3*e^3 + (5*B*b^2 + 3*A*b*c)*d^2*e^
4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^6*e^3 - 2*b*c*d^5*e^4 + b^2*d^4*e^5 + (c^2*d^4*e^5 - 2*b*c*d^3*e^6 + b^2*d^2*e
^7)*x^2 + 2*(c^2*d^5*e^4 - 2*b*c*d^4*e^5 + b^2*d^3*e^6)*x), -1/4*((8*B*c^2*d^5 - 12*B*b*c*d^4*e + 3*B*b^2*d^3*
e^2 + A*b^2*d^2*e^3 + (8*B*c^2*d^3*e^2 - 12*B*b*c*d^2*e^3 + 3*B*b^2*d*e^4 + A*b^2*e^5)*x^2 + 2*(8*B*c^2*d^4*e
- 12*B*b*c*d^3*e^2 + 3*B*b^2*d^2*e^3 + A*b^2*d*e^4)*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)) + 8*(B*c^2*d^6 - 2*B*b*c*d^5*e + B*b^2*d^4*e^2 + (B*c^2*d^4*e^2 - 2*B*b*c*d^3*e^
3 + B*b^2*d^2*e^4)*x^2 + 2*(B*c^2*d^5*e - 2*B*b*c*d^4*e^2 + B*b^2*d^3*e^3)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x
)*sqrt(-c)/(c*x)) + (4*B*c^2*d^5*e - 7*B*b*c*d^4*e^2 + A*b^2*d^2*e^4 + (3*B*b^2 - A*b*c)*d^3*e^3 + (6*B*c^2*d^
4*e^2 - A*b^2*d*e^5 - (11*B*b*c + 2*A*c^2)*d^3*e^3 + (5*B*b^2 + 3*A*b*c)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d
^6*e^3 - 2*b*c*d^5*e^4 + b^2*d^4*e^5 + (c^2*d^4*e^5 - 2*b*c*d^3*e^6 + b^2*d^2*e^7)*x^2 + 2*(c^2*d^5*e^4 - 2*b*
c*d^4*e^5 + b^2*d^3*e^6)*x)]

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giac [B]  time = 0.40, size = 819, normalized size = 3.49 \begin {gather*} -B \sqrt {c} e^{\left (-3\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right ) - \frac {{\left (8 \, B c^{2} d^{3} - 12 \, B b c d^{2} e + 3 \, B b^{2} 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 )}{4 \, {\left (c d^{2} e^{3} - b d e^{4}\right )} \sqrt {-c d^{2} + b d e}} - \frac {16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B c^{\frac {5}{2}} d^{3} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B c^{3} d^{4} - 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b c^{2} d^{3} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c^{3} d^{3} e + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b c^{\frac {5}{2}} d^{4} - 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b c^{\frac {3}{2}} d^{2} e^{2} - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{\frac {5}{2}} d^{2} e^{2} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{2} c^{\frac {3}{2}} d^{3} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b c^{\frac {5}{2}} d^{3} e + 6 \, B b^{2} c^{2} d^{4} - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} c d^{2} e^{2} - 5 \, B b^{3} c d^{3} e - 2 \, A b^{2} c^{2} d^{3} e + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} \sqrt {c} d e^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c^{\frac {3}{2}} d e^{3} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} B b^{3} \sqrt {c} d^{2} e^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} c^{\frac {3}{2}} d^{2} e^{2} + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} c d e^{3} + A b^{3} c d^{2} e^{2} - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} \sqrt {c} e^{4} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} \sqrt {c} d e^{3}}{4 \, {\left (c^{\frac {3}{2}} d^{2} e^{3} - b \sqrt {c} d e^{4}\right )} {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*sqrt(c)*e^(-3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b)) - 1/4*(8*B*c^2*d^3 - 12*B*b*c*d^2*e
+ 3*B*b^2*d*e^2 + A*b^2*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c
*d^2*e^3 - b*d*e^4)*sqrt(-c*d^2 + b*d*e)) - 1/4*(16*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^(5/2)*d^3*e + 24*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^2*B*c^3*d^4 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^2*d^3*e - 8*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^2*A*c^3*d^3*e + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^(5/2)*d^4 - 20*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^3*B*b*c^(3/2)*d^2*e^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(5/2)*d^2*e^2 - 24*(sqrt(c)*x -
sqrt(c*x^2 + b*x))*B*b^2*c^(3/2)*d^3*e - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*c^(5/2)*d^3*e + 6*B*b^2*c^2*d^4
- (sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^2*c*d^2*e^2 - 5*B*b^3*c*d^3*e - 2*A*b^2*c^2*d^3*e + 5*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^3*B*b^2*sqrt(c)*d*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^(3/2)*d*e^3 + 3*(sqrt(c)*x
- sqrt(c*x^2 + b*x))*B*b^3*sqrt(c)*d^2*e^2 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*c^(3/2)*d^2*e^2 + 5*(sqrt
(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c*d*e^3 + A*b^3*c*d^2*e^2 - (sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*sqrt(c)
*e^4 + (sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*sqrt(c)*d*e^3)/((c^(3/2)*d^2*e^3 - b*sqrt(c)*d*e^4)*((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)

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maple [B]  time = 0.07, size = 4316, normalized size = 18.37 \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)^(1/2)/(e*x+d)^3,x)

[Out]

-1/4/e*c^(1/2)/(b*e-c*d)/d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(
x+d/e)/e)^(1/2))*b*A-3/2*B/e^4/(b*e-c*d)*d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/
e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*c^2-1/2/e^
4/(b*e-c*d)^2*d^3/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(
1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*c^3*B-1/2/e^2*c/(b*e-c*d)/(-(b*e-c*d)
*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d
/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*A+1/2/e^3*c^2/(b*e-c*d)*d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c
*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)
^(1/2))/(x+d/e))*A+1/4*e/(b*e-c*d)^2/d^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*b*A
+3/4/e^2/(b*e-c*d)^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e)^(1/2))*c^(3/2)*b*B*d+5/8/e/(b*e-c*d)^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)
/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2*c*A+3/4/
(b*e-c*d)^2/d*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*c*A+1/2/(b*e-c*d)^2/d*c^2*((x+d/e)^2
*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*A+1/4/(b*e-c*d)^2/d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)
+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b^2*A+1/e^3/(b*e-c*d)^2*d^2/(-(b*e-c*d)*d/
e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^
2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*c^2*B-5/8/e^2/(b*e-c*d)^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)
*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1
/2))/(x+d/e))*b^2*c*B*d+2*B/e^3/(b*e-c*d)*d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e
)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*c-1/4*e/(
b*e-c*d)^2/d^2*c*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*b*A-1/e^2/(b*e-c*d)^2*d/(-(b*e-c*
d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)
*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b*c^2*A-1/4/(b*e-c*d)^2/d/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(
b*e-2*c*d)*(x+d/e)/e)^(3/2)*b*B-1/2/(b*e-c*d)^2/d/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^
(3/2)*c*A-1/8/(b*e-c*d)^2/d/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d
)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^3*A+1/2/e/(b*e-c*d)/d/(x+
d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)*A-1/4*e/(b*e-c*d)^2/d^2*((x+d/e)^2*c-(b*e-c*d
)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*A+1/2/e^2/(b*e-c*d)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e
)/e)^(1/2)*c^2*B*d+1/2/e^2/(b*e-c*d)^2*d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2
+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(5/2)*A-1/2/e^3/(b*e-c*d)^2*d^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+
d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(5/2)*B-1/2/e*c/(b*e-c*d)/d*((x+d/e)^2*c-(b*e-c*d)*d/
e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*A-3/4/e/(b*e-c*d)^2*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*
e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(3/2)*b*A-3/2*B/e^3/(b*e-c*d)*d*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/
c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(3/2)-1/2*B/e^2/(b*e-c*d)/(-(b*e-c*d)*d/e
^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2
+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x+d/e))*b^2+1/8/e/(b*e-c*d)^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2*(b*e-c*d)*d/e^2+
(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))/(x
+d/e))*b^3*B-3/4/e/(b*e-c*d)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b*c*B-1/2/e/(b*e-c*d)
^2*c^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*B-1/4/e/(b*e-c*d)^2*ln(((x+d/e)*c+1/2*(b*e-
2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b^2*B-B/e/(b*e-c*d)/d*((x
+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b+5/4*B/e^2/(b*e-c*d)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/
c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2))*c^(1/2)*b+B/e/(b*e-c*d)/d/(x+d/e)*((x+d/e)^
2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/2)+1/2/e/(b*e-c*d)^2/(x+d/e)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2
*c*d)*(x+d/e)/e)^(3/2)*c*B-1/2/e^2/(b*e-c*d)/(x+d/e)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(3/
2)*B-1/2/e/(b*e-c*d)^2*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c^2*A-B/e*c/(b*e-c*d)/d*((x+d
/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x+1/2/e^3/(b*e-c*d)^2*d^2/(-(b*e-c*d)*d/e^2)^(1/2)*ln((-2
*(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e+2*(-(b*e-c*d)*d/e^2)^(1/2)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+
d/e)/e)^(1/2))/(x+d/e))*c^3*A+1/4/(b*e-c*d)^2/d*c*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*x*
b*B+1/2/e^2*c^(3/2)/(b*e-c*d)*ln(((x+d/e)*c+1/2*(b*e-2*c*d)/e)/c^(1/2)+((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d
)*(x+d/e)/e)^(1/2))*A+1/4/(b*e-c*d)^2/d*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*b^2*B+3/2*B/
e^2/(b*e-c*d)*((x+d/e)^2*c-(b*e-c*d)*d/e^2+(b*e-2*c*d)*(x+d/e)/e)^(1/2)*c

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

[Out]

int(((b*x + c*x^2)^(1/2)*(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 {\sqrt {x \left (b + c x\right )} \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)**(1/2)/(e*x+d)**3,x)

[Out]

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

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