3.1175 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{d+e x} \, dx\)

Optimal. Leaf size=392 \[ \frac{\sqrt{b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (8 b^2 c d e^2+3 b^3 e^3-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{64 c^2 e^4}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (8 A c e \left (6 b^2 c d e^2+b^3 e^3-24 b c^2 d^2 e+16 c^3 d^3\right )-B \left (48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{64 c^{5/2} e^5}-\frac{d^{3/2} (B d-A e) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^5}-\frac{\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2} \]

[Out]

((8*A*c*e*(8*c^2*d^2 - 10*b*c*d*e + b^2*e^2) - B*(64*c^3*d^3 - 80*b*c^2*d^2*e + 8*b^2*c*d*e^2 + 3*b^3*e^3) - 2
*c*e*(8*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c^2*e^4) - ((8
*B*c*d - 3*b*B*e - 8*A*c*e - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(24*c*e^2) - ((8*A*c*e*(16*c^3*d^3 - 24*b*c^2*d^2
*e + 6*b^2*c*d*e^2 + b^3*e^3) - B*(128*c^4*d^4 - 192*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 + 3*b^4*
e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(5/2)*e^5) - (d^(3/2)*(B*d - A*e)*(c*d - b*e)^(3/2)*ArcTan
h[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e^5

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Rubi [A]  time = 0.590233, antiderivative size = 392, normalized size of antiderivative = 1., 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 = {814, 843, 620, 206, 724} \[ \frac{\sqrt{b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (8 b^2 c d e^2+3 b^3 e^3-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{64 c^2 e^4}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (8 A c e \left (6 b^2 c d e^2+b^3 e^3-24 b c^2 d^2 e+16 c^3 d^3\right )-B \left (48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4-192 b c^3 d^3 e+128 c^4 d^4\right )\right )}{64 c^{5/2} e^5}-\frac{d^{3/2} (B d-A e) (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^5}-\frac{\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*A*c*e*(8*c^2*d^2 - 10*b*c*d*e + b^2*e^2) - B*(64*c^3*d^3 - 80*b*c^2*d^2*e + 8*b^2*c*d*e^2 + 3*b^3*e^3) - 2
*c*e*(8*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(64*c^2*e^4) - ((8
*B*c*d - 3*b*B*e - 8*A*c*e - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(24*c*e^2) - ((8*A*c*e*(16*c^3*d^3 - 24*b*c^2*d^2
*e + 6*b^2*c*d*e^2 + b^3*e^3) - B*(128*c^4*d^4 - 192*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 + 3*b^4*
e^4))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(64*c^(5/2)*e^5) - (d^(3/2)*(B*d - A*e)*(c*d - b*e)^(3/2)*ArcTan
h[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e^5

Rule 814

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

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 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 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]

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx &=-\frac{(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}-\frac{\int \frac{\left (-\frac{1}{2} b d (8 B c d-3 b B e-8 A c e)+\frac{1}{2} \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{d+e x} \, dx}{8 c e^2}\\ &=\frac{\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 c^2 e^4}-\frac{(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}+\frac{\int \frac{-\frac{1}{4} b d \left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )\right )-\frac{1}{4} \left (4 b c d e (2 c d-b e) (8 B c d-3 b B e-8 A c e)+\left (8 c^2 d^2-4 b c d e-b^2 e^2\right ) \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right )\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{32 c^2 e^4}\\ &=\frac{\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 c^2 e^4}-\frac{(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}-\frac{\left (d^2 (B d-A e) (c d-b e)^2\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{e^5}-\frac{\left (8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )-B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c^2 e^5}\\ &=\frac{\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 c^2 e^4}-\frac{(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}+\frac{\left (2 d^2 (B d-A e) (c d-b e)^2\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 )}{e^5}-\frac{\left (8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )-B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c^2 e^5}\\ &=\frac{\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 c^2 e^4}-\frac{(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}-\frac{\left (8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )-B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2} e^5}-\frac{d^{3/2} (B d-A e) (c d-b e)^{3/2} \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 )}{e^5}\\ \end{align*}

Mathematica [A]  time = 2.04492, size = 387, normalized size = 0.99 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (-e \sqrt{x} \left (B \left (-6 b^2 c e^2 (e x-4 d)+9 b^3 e^3-8 b c^2 e \left (30 d^2-14 d e x+9 e^2 x^2\right )+16 c^3 \left (-6 d^2 e x+12 d^3+4 d e^2 x^2-3 e^3 x^3\right )\right )-8 A c e \left (3 b^2 e^2+2 b c e (7 e x-15 d)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )-\frac{384 c^2 d^{3/2} (B d-A e) (b e-c d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x}}\right )+\frac{3 \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right ) \left (B \left (48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4-192 b c^3 d^3 e+128 c^4 d^4\right )-8 A c e \left (6 b^2 c d e^2+b^3 e^3-24 b c^2 d^2 e+16 c^3 d^3\right )\right )}{\sqrt{b} \sqrt{\frac{c x}{b}+1}}\right )}{192 c^{5/2} e^5 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*((3*(-8*A*c*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3) + B*(128*c^4*d^4 - 19
2*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 + 8*b^3*c*d*e^3 + 3*b^4*e^4))*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*
Sqrt[1 + (c*x)/b]) + Sqrt[c]*(-(e*Sqrt[x]*(-8*A*c*e*(3*b^2*e^2 + 2*b*c*e*(-15*d + 7*e*x) + 4*c^2*(6*d^2 - 3*d*
e*x + 2*e^2*x^2)) + B*(9*b^3*e^3 - 6*b^2*c*e^2*(-4*d + e*x) - 8*b*c^2*e*(30*d^2 - 14*d*e*x + 9*e^2*x^2) + 16*c
^3*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3*x^3)))) - (384*c^2*d^(3/2)*(B*d - A*e)*(-(c*d) + b*e)^(3/2)*ArcTa
n[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/Sqrt[b + c*x])))/(192*c^(5/2)*e^5*Sqrt[x])

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Maple [B]  time = 0.008, size = 2334, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

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),x)

[Out]

1/3/e*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*A-3/64*B/e*b^3/c^2*(c*x^2+b*x)^(1/2)+3/128*B/e
*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))-2/e^5*d^4/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e
^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))
/(x+d/e))*b*c*B+2/e^4*d^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d
)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b*c*A+1/e^6*d^5/(-d*(b*e-c*d)
/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*c^2*B-1/2/e^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1
/2)*x*c*d*A+1/16/e^2/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*
e-c*d)/e^2)^(1/2))*b^3*B*d-1/8/e^2/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b^2*B*d-1/4/e^2
*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*b*B*d+3/8/e^3*d^2*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c
)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/c^(1/2)*b^2*B+3/2/e^3*d^2*ln((1/2*(b*e-2*
c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b*A-3/32*B/e*b^2/
c*(c*x^2+b*x)^(1/2)*x-3/2/e^4*d^3*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-
d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b*B+1/2/e^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c*d^2*
B-3/8/e^2*d*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)
)/c^(1/2)*b^2*A-1/e^3*d^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d
)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2*A+1/e^4*d^3/(-d*(b*e-c*d)
/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b^2*B-1/e^5*d^4/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e
-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e
))*c^2*A-1/3/e^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*B*d+1/4*B/e*(c*x^2+b*x)^(3/2)*x+1/e
^5*d^4*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(
3/2)*B+5/4/e^3*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*d^2*B-1/16/e/c^(3/2)*ln((1/2*(b*e-2
*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*b^3*A+1/4/e*((x+d/e)^2*c
+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*b*A+1/8/e/c*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2
)^(1/2)*b^2*A+1/e^3*d^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c*A-1/e^4*d^3*((x+d/e)^2*c+(
b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c*B+1/8*B/e/c*(c*x^2+b*x)^(3/2)*b-5/4/e^2*((x+d/e)^2*c+(b*e-2*c*d)
/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*d*A-1/e^4*d^3*ln((1/2*(b*e-2*c*d)/e+(x+d/e)*c)/c^(1/2)+((x+d/e)^2*c+(b*e-2
*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*A

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

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 153.846, size = 3680, normalized size = 9.39 \begin{align*} \text{result too large to display} \end{align*}

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),x, algorithm="fricas")

[Out]

[-1/384*(3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c -
6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 384*(B
*c^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*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*(48*B*c^4*e^4*x^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)*
d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^2)*e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3
+ 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)
*sqrt(c*x^2 + b*x))/(c^3*e^5), -1/384*(768*(B*c^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*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*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2
*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e
^4)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(48*B*c^4*e^4*x^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c
^3 + 4*A*c^4)*d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^2)*e^4 - 8*(8*B*c^4*d*e^3
- (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*
b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^3*e^5), -1/192*(3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B
*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*sqrt(-c)*arctan(s
qrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + 192*(B*c^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*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)) - (48*B*c^4*e^4*x^3 - 192*
B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)*d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^
2)*e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e
^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^3*e^5), -1/192*(384*(B*c^4*d^3 + A*b*c^3*d*e^2 -
(B*b*c^3 + A*c^4)*d^2*e)*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*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b
^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - (48*B*c^4*e^4*x
^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)*d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8
*A*b^2*c^2)*e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A
*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)*sqrt(c*x^2 + b*x))/(c^3*e^5)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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),x, algorithm="giac")

[Out]

Exception raised: TypeError