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

Optimal. Leaf size=305 \[ \frac {\sqrt {b x+c x^2} \left (2 c e x \left (40 A c e (2 c d-b e)+B \left (35 b^2 e^2-64 b c d e+24 c^2 d^2\right )\right )+8 A c e \left (15 b^2 e^2-54 b c d e+64 c^2 d^2\right )+B \left (-105 b^3 e^3+360 b^2 c d e^2-376 b c^2 d^2 e+96 c^3 d^3\right )\right )}{192 c^4}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (-40 b^3 c e^2 (A e+3 B d)+144 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{64 c^{9/2}}+\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c} \]

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Rubi [A]  time = 0.44, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {832, 779, 620, 206} \begin {gather*} \frac {\sqrt {b x+c x^2} \left (2 c e x \left (40 A c e (2 c d-b e)+B \left (35 b^2 e^2-64 b c d e+24 c^2 d^2\right )\right )+8 A c e \left (15 b^2 e^2-54 b c d e+64 c^2 d^2\right )+B \left (360 b^2 c d e^2-105 b^3 e^3-376 b c^2 d^2 e+96 c^3 d^3\right )\right )}{192 c^4}+\frac {\tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (144 b^2 c^2 d e (A e+B d)-40 b^3 c e^2 (A e+3 B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{64 c^{9/2}}+\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{24 c^2}+\frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*B*c*d - 7*b*B*e + 8*A*c*e)*(d + e*x)^2*Sqrt[b*x + c*x^2])/(24*c^2) + (B*(d + e*x)^3*Sqrt[b*x + c*x^2])/(4*
c) + ((8*A*c*e*(64*c^2*d^2 - 54*b*c*d*e + 15*b^2*e^2) + B*(96*c^3*d^3 - 376*b*c^2*d^2*e + 360*b^2*c*d*e^2 - 10
5*b^3*e^3) + 2*c*e*(40*A*c*e*(2*c*d - b*e) + B*(24*c^2*d^2 - 64*b*c*d*e + 35*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(
192*c^4) + ((128*A*c^4*d^3 + 35*b^4*B*e^3 + 144*b^2*c^2*d*e*(B*d + A*e) - 40*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^
3*d^2*(B*d + 3*A*e))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*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 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 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 {b x+c x^2}} \, dx &=\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\int \frac {(d+e x)^2 \left (-\frac {1}{2} (b B-8 A c) d+\frac {1}{2} (6 B c d-7 b B e+8 A c e) x\right )}{\sqrt {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 {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\int \frac {(d+e x) \left (-\frac {1}{4} d \left (12 b B c d-48 A c^2 d-7 b^2 B e+8 A b c e\right )+\frac {1}{4} \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right )}{\sqrt {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 {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{192 c^4}+\frac {\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \int \frac {1}{\sqrt {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 {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{192 c^4}+\frac {\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {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 {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{192 c^4}+\frac {\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.79, size = 278, normalized size = 0.91 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (8 A c e \left (15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+B \left (-105 b^3 e^3+10 b^2 c e^2 (36 d+7 e x)-8 b c^2 e \left (54 d^2+30 d e x+7 e^2 x^2\right )+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 )+\frac {3 \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ) \left (-40 b^3 c e^2 (A e+3 B d)+144 b^2 c^2 d e (A e+B d)-64 b c^3 d^2 (3 A e+B d)+128 A c^4 d^3+35 b^4 B e^3\right )}{\sqrt {b} \sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{192 c^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(8*A*c*e*(15*b^2*e^2 - 2*b*c*e*(27*d + 5*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^2*c*e^2*(36*d + 7*e*x) - 8*b*c^2*e*(54*d^2 + 30*d*e*x + 7*e^2*x^2) + 48*c^3*(4*d^
3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))) + (3*(128*A*c^4*d^3 + 35*b^4*B*e^3 + 144*b^2*c^2*d*e*(B*d + A*e) - 40
*b^3*c*e^2*(3*B*d + A*e) - 64*b*c^3*d^2*(B*d + 3*A*e))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[x]*Sq
rt[1 + (c*x)/b])))/(192*c^(9/2))

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IntegrateAlgebraic [A]  time = 1.22, size = 322, normalized size = 1.06 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (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}+\frac {\log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right ) \left (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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[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 - 105*b^3*B*e^3 + 120*A*b^2*c*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 + 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 - 128*A*c^4*d^3 - 144*b^2*B*c^2*d^2*e + 192*A*b*c^3*d^2*e + 120*b^3*B*c*d*e^2 - 14
4*A*b^2*c^2*d*e^2 - 35*b^4*B*e^3 + 40*A*b^3*c*e^3)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[b*x + c*x^2]])/(128*c^(9/2))

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fricas [A]  time = 0.45, size = 621, normalized size = 2.04 \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 \, A b c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\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 + 72 \, {\left (5 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} d e^{2} - 15 \, {\left (7 \, B b^{3} c - 8 \, A b^{2} 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} + 5 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{384 \, 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 \, A b c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\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 + 72 \, {\left (5 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} d e^{2} - 15 \, {\left (7 \, B b^{3} c - 8 \, A b^{2} 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} + 5 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{192 \, 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)^(1/2),x, algorithm="fricas")

[Out]

[1/384*(3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*A*b*c^3)*d^2*e + 24*(5*B*b^3*c - 6*A*b^2*c^2)*d*e^
2 - 5*(7*B*b^4 - 8*A*b^3*c)*e^3)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(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 + 72*(5*B*b^2*c^2 - 6*A*b*c^3)*d*e^2 - 15*(7*B*b^3*c - 8*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 + 5*(7*B*b^2*c^2 - 8*A*b*c^3)*e^3)*x)*sqrt(c*x^2 + b*x))/c^5, 1/192*(3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48
*(3*B*b^2*c^2 - 4*A*b*c^3)*d^2*e + 24*(5*B*b^3*c - 6*A*b^2*c^2)*d*e^2 - 5*(7*B*b^4 - 8*A*b^3*c)*e^3)*sqrt(-c)*
arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (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
+ 72*(5*B*b^2*c^2 - 6*A*b*c^3)*d*e^2 - 15*(7*B*b^3*c - 8*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 + 5*(7*B*b^2*c^2 - 8*A*b*c^3)*e^3)*x)*sq
rt(c*x^2 + b*x))/c^5]

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giac [A]  time = 0.33, size = 311, normalized size = 1.02 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x} {\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} - 40 \, A b c^{2} e^{3}}{c^{4}}\right )} x + \frac {3 \, {\left (64 \, B c^{3} d^{3} - 144 \, B b c^{2} d^{2} e + 192 \, A c^{3} d^{2} e + 120 \, B b^{2} c d e^{2} - 144 \, A b c^{2} d e^{2} - 35 \, B b^{3} e^{3} + 40 \, A b^{2} c e^{3}\right )}}{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 \, A b c^{3} d^{2} e + 120 \, B b^{3} c d e^{2} - 144 \, A b^{2} c^{2} d e^{2} - 35 \, B b^{4} e^{3} + 40 \, A b^{3} c e^{3}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\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)^(1/2),x, algorithm="giac")

[Out]

1/192*sqrt(c*x^2 + b*x)*(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 - 40*A*b*c^2*e^3)/c^4)*x + 3*(64*B*c^3*d^3 - 14
4*B*b*c^2*d^2*e + 192*A*c^3*d^2*e + 120*B*b^2*c*d*e^2 - 144*A*b*c^2*d*e^2 - 35*B*b^3*e^3 + 40*A*b^2*c*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*A*b*c^3*d^2*e + 120*B*b^3*c*d*e^2 - 144*
A*b^2*c^2*d*e^2 - 35*B*b^4*e^3 + 40*A*b^3*c*e^3)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(9
/2)

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maple [B]  time = 0.06, size = 646, normalized size = 2.12 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x}\, B \,e^{3} x^{3}}{4 c}+\frac {\sqrt {c \,x^{2}+b x}\, A \,e^{3} x^{2}}{3 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, B b \,e^{3} x^{2}}{24 c^{2}}+\frac {\sqrt {c \,x^{2}+b x}\, B d \,e^{2} x^{2}}{c}-\frac {5 A \,b^{3} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\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}\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}\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}\right )}{\sqrt {c}}+\frac {35 B \,b^{4} e^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\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}\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}\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}\right )}{2 c^{\frac {3}{2}}}-\frac {5 \sqrt {c \,x^{2}+b x}\, A b \,e^{3} x}{12 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A d \,e^{2} x}{2 c}+\frac {35 \sqrt {c \,x^{2}+b x}\, B \,b^{2} e^{3} x}{96 c^{3}}-\frac {5 \sqrt {c \,x^{2}+b x}\, B b d \,e^{2} x}{4 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, B \,d^{2} e x}{2 c}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{2} e^{3}}{8 c^{3}}-\frac {9 \sqrt {c \,x^{2}+b x}\, A b d \,e^{2}}{4 c^{2}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,d^{2} e}{c}-\frac {35 \sqrt {c \,x^{2}+b x}\, B \,b^{3} e^{3}}{64 c^{4}}+\frac {15 \sqrt {c \,x^{2}+b x}\, B \,b^{2} d \,e^{2}}{8 c^{3}}-\frac {9 \sqrt {c \,x^{2}+b x}\, B b \,d^{2} e}{4 c^{2}}+\frac {\sqrt {c \,x^{2}+b x}\, 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)^(1/2),x)

[Out]

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

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maxima [A]  time = 0.59, size = 474, normalized size = 1.55 \begin {gather*} \frac {\sqrt {c x^{2} + b x} B e^{3} x^{3}}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b e^{3} x^{2}}{24 \, c^{2}} + \frac {35 \, \sqrt {c x^{2} + b x} B b^{2} e^{3} x}{96 \, c^{3}} + \frac {A d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} + \frac {35 \, B b^{4} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {9}{2}}} - \frac {35 \, \sqrt {c x^{2} + b x} B b^{3} e^{3}}{64 \, c^{4}} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} x^{2}}{3 \, c} - \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b x}{12 \, c^{2}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} x}{2 \, c} - \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} + \frac {9 \, {\left (B d^{2} e + A d e^{2}\right )} b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b^{2}}{8 \, c^{3}} - \frac {9 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} b}{4 \, c^{2}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {c x^{2} + b x}}{c} \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)^(1/2),x, algorithm="maxima")

[Out]

1/4*sqrt(c*x^2 + b*x)*B*e^3*x^3/c - 7/24*sqrt(c*x^2 + b*x)*B*b*e^3*x^2/c^2 + 35/96*sqrt(c*x^2 + b*x)*B*b^2*e^3
*x/c^3 + A*d^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/sqrt(c) + 35/128*B*b^4*e^3*log(2*c*x + b + 2*sqrt(
c*x^2 + b*x)*sqrt(c))/c^(9/2) - 35/64*sqrt(c*x^2 + b*x)*B*b^3*e^3/c^4 + 1/3*(3*B*d*e^2 + A*e^3)*sqrt(c*x^2 + b
*x)*x^2/c - 5/12*(3*B*d*e^2 + A*e^3)*sqrt(c*x^2 + b*x)*b*x/c^2 + 3/2*(B*d^2*e + A*d*e^2)*sqrt(c*x^2 + b*x)*x/c
- 5/16*(3*B*d*e^2 + A*e^3)*b^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(7/2) + 9/8*(B*d^2*e + A*d*e^2)
*b^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) - 1/2*(B*d^3 + 3*A*d^2*e)*b*log(2*c*x + b + 2*sqrt(c
*x^2 + b*x)*sqrt(c))/c^(3/2) + 5/8*(3*B*d*e^2 + A*e^3)*sqrt(c*x^2 + b*x)*b^2/c^3 - 9/4*(B*d^2*e + A*d*e^2)*sqr
t(c*x^2 + b*x)*b/c^2 + (B*d^3 + 3*A*d^2*e)*sqrt(c*x^2 + b*x)/c

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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}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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