3.1179 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=421 \[ -\frac{\sqrt{b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (98 b^2 c d e^2-5 b^3 e^3-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (40 b^2 c d e^2+5 b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{64 d^2 e^4 (d+e x)^2 (c d-b e)^2}+\frac{\left (3 A b^4 e^5-B d \left (240 b^2 c^2 d^2 e^2-40 b^3 c d e^3-5 b^4 e^4-320 b c^3 d^3 e+128 c^4 d^4\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{5/2} e^5 (c d-b e)^{5/2}}+\frac{\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}+\frac{2 B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^5} \]

[Out]

-((d*(3*A*b^3*e^4 + B*d*(64*c^3*d^3 - 112*b*c^2*d^2*e + 40*b^2*c*d*e^2 + 5*b^3*e^3)) + e*(3*A*b^2*e^3*(2*c*d -
b*e) + B*d*(96*c^3*d^3 - 192*b*c^2*d^2*e + 98*b^2*c*d*e^2 - 5*b^3*e^3))*x)*Sqrt[b*x + c*x^2])/(64*d^2*e^4*(c*
d - b*e)^2*(d + e*x)^2) + ((d*(3*A*b*e^2 - B*d*(8*c*d - 5*b*e)) - e*(B*d*(14*c*d - 11*b*e) - 3*A*e*(2*c*d - b*
e))*x)*(b*x + c*x^2)^(3/2))/(24*d*e^2*(c*d - b*e)*(d + e*x)^4) + (2*B*c^(3/2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c
*x^2]])/e^5 + ((3*A*b^4*e^5 - B*d*(128*c^4*d^4 - 320*b*c^3*d^3*e + 240*b^2*c^2*d^2*e^2 - 40*b^3*c*d*e^3 - 5*b^
4*e^4))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(128*d^(5/2)*e^5*(c*d
- b*e)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.538537, antiderivative size = 421, 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 = {810, 843, 620, 206, 724} \[ -\frac{\sqrt{b x+c x^2} \left (e x \left (3 A b^2 e^3 (2 c d-b e)+B d \left (98 b^2 c d e^2-5 b^3 e^3-192 b c^2 d^2 e+96 c^3 d^3\right )\right )+d \left (3 A b^3 e^4+B d \left (40 b^2 c d e^2+5 b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )\right )\right )}{64 d^2 e^4 (d+e x)^2 (c d-b e)^2}+\frac{\left (3 A b^4 e^5-B d \left (240 b^2 c^2 d^2 e^2-40 b^3 c d e^3-5 b^4 e^4-320 b c^3 d^3 e+128 c^4 d^4\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{128 d^{5/2} e^5 (c d-b e)^{5/2}}+\frac{\left (b x+c x^2\right )^{3/2} \left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e x (B d (14 c d-11 b e)-3 A e (2 c d-b e))\right )}{24 d e^2 (d+e x)^4 (c d-b e)}+\frac{2 B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^5} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^5,x]

[Out]

-((d*(3*A*b^3*e^4 + B*d*(64*c^3*d^3 - 112*b*c^2*d^2*e + 40*b^2*c*d*e^2 + 5*b^3*e^3)) + e*(3*A*b^2*e^3*(2*c*d -
b*e) + B*d*(96*c^3*d^3 - 192*b*c^2*d^2*e + 98*b^2*c*d*e^2 - 5*b^3*e^3))*x)*Sqrt[b*x + c*x^2])/(64*d^2*e^4*(c*
d - b*e)^2*(d + e*x)^2) + ((d*(3*A*b*e^2 - B*d*(8*c*d - 5*b*e)) - e*(B*d*(14*c*d - 11*b*e) - 3*A*e*(2*c*d - b*
e))*x)*(b*x + c*x^2)^(3/2))/(24*d*e^2*(c*d - b*e)*(d + e*x)^4) + (2*B*c^(3/2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c
*x^2]])/e^5 + ((3*A*b^4*e^5 - B*d*(128*c^4*d^4 - 320*b*c^3*d^3*e + 240*b^2*c^2*d^2*e^2 - 40*b^3*c*d*e^3 - 5*b^
4*e^4))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(128*d^(5/2)*e^5*(c*d
- b*e)^(5/2))

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]

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)^5} \, dx &=\frac{\left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e (B d (14 c d-11 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{24 d e^2 (c d-b e) (d+e x)^4}-\frac{\int \frac{\left (\frac{1}{2} b \left (3 A b e^2-B d (8 c d-5 b e)\right )-8 B c d (c d-b e) x\right ) \sqrt{b x+c x^2}}{(d+e x)^3} \, dx}{8 d e^2 (c d-b e)}\\ &=-\frac{\left (d \left (3 A b^3 e^4+B d \left (64 c^3 d^3-112 b c^2 d^2 e+40 b^2 c d e^2+5 b^3 e^3\right )\right )+e \left (3 A b^2 e^3 (2 c d-b e)+B d \left (96 c^3 d^3-192 b c^2 d^2 e+98 b^2 c d e^2-5 b^3 e^3\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 d^2 e^4 (c d-b e)^2 (d+e x)^2}+\frac{\left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e (B d (14 c d-11 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{24 d e^2 (c d-b e) (d+e x)^4}+\frac{\int \frac{\frac{1}{4} b \left (3 A b^3 e^4+B d \left (64 c^3 d^3-112 b c^2 d^2 e+40 b^2 c d e^2+5 b^3 e^3\right )\right )+32 B c^2 d^2 (c d-b e)^2 x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{32 d^2 e^4 (c d-b e)^2}\\ &=-\frac{\left (d \left (3 A b^3 e^4+B d \left (64 c^3 d^3-112 b c^2 d^2 e+40 b^2 c d e^2+5 b^3 e^3\right )\right )+e \left (3 A b^2 e^3 (2 c d-b e)+B d \left (96 c^3 d^3-192 b c^2 d^2 e+98 b^2 c d e^2-5 b^3 e^3\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 d^2 e^4 (c d-b e)^2 (d+e x)^2}+\frac{\left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e (B d (14 c d-11 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{24 d e^2 (c d-b e) (d+e x)^4}+\frac{\left (B c^2\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{e^5}+\frac{\left (3 A b^4 e^5-B d \left (128 c^4 d^4-320 b c^3 d^3 e+240 b^2 c^2 d^2 e^2-40 b^3 c d e^3-5 b^4 e^4\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{128 d^2 e^5 (c d-b e)^2}\\ &=-\frac{\left (d \left (3 A b^3 e^4+B d \left (64 c^3 d^3-112 b c^2 d^2 e+40 b^2 c d e^2+5 b^3 e^3\right )\right )+e \left (3 A b^2 e^3 (2 c d-b e)+B d \left (96 c^3 d^3-192 b c^2 d^2 e+98 b^2 c d e^2-5 b^3 e^3\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 d^2 e^4 (c d-b e)^2 (d+e x)^2}+\frac{\left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e (B d (14 c d-11 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{24 d e^2 (c d-b e) (d+e x)^4}+\frac{\left (2 B c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{e^5}-\frac{\left (3 A b^4 e^5-B d \left (128 c^4 d^4-320 b c^3 d^3 e+240 b^2 c^2 d^2 e^2-40 b^3 c d e^3-5 b^4 e^4\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 )}{64 d^2 e^5 (c d-b e)^2}\\ &=-\frac{\left (d \left (3 A b^3 e^4+B d \left (64 c^3 d^3-112 b c^2 d^2 e+40 b^2 c d e^2+5 b^3 e^3\right )\right )+e \left (3 A b^2 e^3 (2 c d-b e)+B d \left (96 c^3 d^3-192 b c^2 d^2 e+98 b^2 c d e^2-5 b^3 e^3\right )\right ) x\right ) \sqrt{b x+c x^2}}{64 d^2 e^4 (c d-b e)^2 (d+e x)^2}+\frac{\left (d \left (3 A b e^2-B d (8 c d-5 b e)\right )-e (B d (14 c d-11 b e)-3 A e (2 c d-b e)) x\right ) \left (b x+c x^2\right )^{3/2}}{24 d e^2 (c d-b e) (d+e x)^4}+\frac{2 B c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{e^5}+\frac{\left (3 A b^4 e^5-B d \left (128 c^4 d^4-320 b c^3 d^3 e+240 b^2 c^2 d^2 e^2-40 b^3 c d e^3-5 b^4 e^4\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 )}{128 d^{5/2} e^5 (c d-b e)^{5/2}}\\ \end{align*}

Mathematica [B]  time = 6.20718, size = 2000, normalized size = 4.75 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^5,x]

[Out]

((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(3/2))/(4*d*(-(c*d) + b*e)*(d + e*x)^4) + ((x*(b + c*x))^(3/2)*(((-(
c*d*(B*d - A*e)) + (e*(5*b*B*d - 8*A*c*d + 3*A*b*e))/2)*x^(5/2)*(b + c*x)^(5/2))/(3*d*(-(c*d) + b*e)*(d + e*x)
^3) + ((((e*(48*A*c^2*d^2 + b^2*e*(5*B*d + 3*A*e) - 4*b*c*d*(5*B*d + 9*A*e)))/4 - c*d*(B*d*(2*c*d - 5*b*e) + 3
*A*e*(2*c*d - b*e)))*x^(5/2)*(b + c*x)^(5/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2) + ((((e*(-192*A*c^3*d^3 - b^3*e
^2*(5*B*d + 3*A*e) + 24*b*c^2*d^2*(5*B*d + 9*A*e) - 4*b^2*c*d*e*(25*B*d + 9*A*e)))/8 + (3*c*d*(3*A*e*(8*c^2*d^
2 - 8*b*c*d*e + b^2*e^2) - B*(8*c^2*d^3 - 5*b^2*d*e^2)))/4)*x^(5/2)*(b + c*x)^(5/2))/(d*(-(c*d) + b*e)*(d + e*
x)) + ((-(c*d*(-192*A*c^3*d^3 - b^3*e^2*(5*B*d + 3*A*e) + 24*b*c^2*d^2*(5*B*d + 9*A*e) - 4*b^2*c*d*e*(25*B*d +
9*A*e)))/8 + (b*e*(-192*A*c^3*d^3 - b^3*e^2*(5*B*d + 3*A*e) + 24*b*c^2*d^2*(5*B*d + 9*A*e) - 4*b^2*c*d*e*(25*
B*d + 9*A*e)))/8 - (5*b*((e*(-192*A*c^3*d^3 - b^3*e^2*(5*B*d + 3*A*e) + 24*b*c^2*d^2*(5*B*d + 9*A*e) - 4*b^2*c
*d*e*(25*B*d + 9*A*e)))/8 + (3*c*d*(3*A*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - B*(8*c^2*d^3 - 5*b^2*d*e^2)))/4)
)/2)*((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])/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*S
qrt[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*(c*d - b*e)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*
x])])/(Sqrt[d]*e*Sqrt[-(c*d) + b*e])))/e))/e))/e) - 4*c*((e*(-192*A*c^3*d^3 - b^3*e^2*(5*B*d + 3*A*e) + 24*b*c
^2*d^2*(5*B*d + 9*A*e) - 4*b^2*c*d*e*(25*B*d + 9*A*e)))/8 + (3*c*d*(3*A*e*(8*c^2*d^2 - 8*b*c*d*e + b^2*e^2) -
B*(8*c^2*d^3 - 5*b^2*d*e^2)))/4)*((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])/S
qrt[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]*ArcSi
nh[(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*S
qrt[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[(Sqr
t[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*(c*d - b*e)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(Sqrt[d]*e*Sqrt[-(c*d) + b*
e])))/e))/e))/e))/e))/(d*(-(c*d) + b*e)))/(2*d*(-(c*d) + b*e)))/(3*d*(-(c*d) + b*e))))/(4*d*(-(c*d) + b*e)*x^(
3/2)*(b + c*x)^(3/2))

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Maple [B]  time = 0.022, size = 16396, normalized size = 39. \begin{align*} \text{output 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)^5,x)

[Out]

result too large to display

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 114.825, size = 12124, normalized size = 28.8 \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)^5,x, algorithm="fricas")

[Out]

[1/384*(384*(B*c^4*d^10 - 3*B*b*c^3*d^9*e + 3*B*b^2*c^2*d^8*e^2 - B*b^3*c*d^7*e^3 + (B*c^4*d^6*e^4 - 3*B*b*c^3
*d^5*e^5 + 3*B*b^2*c^2*d^4*e^6 - B*b^3*c*d^3*e^7)*x^4 + 4*(B*c^4*d^7*e^3 - 3*B*b*c^3*d^6*e^4 + 3*B*b^2*c^2*d^5
*e^5 - B*b^3*c*d^4*e^6)*x^3 + 6*(B*c^4*d^8*e^2 - 3*B*b*c^3*d^7*e^3 + 3*B*b^2*c^2*d^6*e^4 - B*b^3*c*d^5*e^5)*x^
2 + 4*(B*c^4*d^9*e - 3*B*b*c^3*d^8*e^2 + 3*B*b^2*c^2*d^7*e^3 - B*b^3*c*d^6*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*s
qrt(c*x^2 + b*x)*sqrt(c)) - 3*(128*B*c^4*d^9 - 320*B*b*c^3*d^8*e + 240*B*b^2*c^2*d^7*e^2 - 40*B*b^3*c*d^6*e^3
- 5*B*b^4*d^5*e^4 - 3*A*b^4*d^4*e^5 + (128*B*c^4*d^5*e^4 - 320*B*b*c^3*d^4*e^5 + 240*B*b^2*c^2*d^3*e^6 - 40*B*
b^3*c*d^2*e^7 - 5*B*b^4*d*e^8 - 3*A*b^4*e^9)*x^4 + 4*(128*B*c^4*d^6*e^3 - 320*B*b*c^3*d^5*e^4 + 240*B*b^2*c^2*
d^4*e^5 - 40*B*b^3*c*d^3*e^6 - 5*B*b^4*d^2*e^7 - 3*A*b^4*d*e^8)*x^3 + 6*(128*B*c^4*d^7*e^2 - 320*B*b*c^3*d^6*e
^3 + 240*B*b^2*c^2*d^5*e^4 - 40*B*b^3*c*d^4*e^5 - 5*B*b^4*d^3*e^6 - 3*A*b^4*d^2*e^7)*x^2 + 4*(128*B*c^4*d^8*e
- 320*B*b*c^3*d^7*e^2 + 240*B*b^2*c^2*d^6*e^3 - 40*B*b^3*c*d^5*e^4 - 5*B*b^4*d^4*e^5 - 3*A*b^4*d^3*e^6)*x)*sqr
t(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*(192*B*c
^4*d^9*e - 528*B*b*c^3*d^8*e^2 + 456*B*b^2*c^2*d^7*e^3 - 105*B*b^3*c*d^6*e^4 - 9*A*b^4*d^4*e^6 - 3*(5*B*b^4 -
3*A*b^3*c)*d^5*e^5 + (400*B*c^4*d^6*e^4 + 9*A*b^4*d*e^9 - 24*(49*B*b*c^3 + 2*A*c^4)*d^5*e^5 + 6*(193*B*b^2*c^2
+ 20*A*b*c^3)*d^4*e^6 - (397*B*b^3*c + 78*A*b^2*c^2)*d^3*e^7 + 3*(5*B*b^4 - A*b^3*c)*d^2*e^8)*x^3 + (832*B*c^
4*d^7*e^3 - 2312*B*b*c^3*d^6*e^4 + 33*A*b^4*d^2*e^8 + 12*(169*B*b^2*c^2 - 6*A*b*c^3)*d^5*e^5 - (475*B*b^3*c -
204*A*b^2*c^2)*d^4*e^6 - (73*B*b^4 + 165*A*b^3*c)*d^3*e^7)*x^2 + (672*B*c^4*d^8*e^2 - 1856*B*b*c^3*d^7*e^3 + 1
614*B*b^2*c^2*d^6*e^4 - 33*A*b^4*d^3*e^7 - 3*(125*B*b^3*c + 2*A*b^2*c^2)*d^5*e^5 - (55*B*b^4 - 39*A*b^3*c)*d^4
*e^6)*x)*sqrt(c*x^2 + b*x))/(c^3*d^10*e^5 - 3*b*c^2*d^9*e^6 + 3*b^2*c*d^8*e^7 - b^3*d^7*e^8 + (c^3*d^6*e^9 - 3
*b*c^2*d^5*e^10 + 3*b^2*c*d^4*e^11 - b^3*d^3*e^12)*x^4 + 4*(c^3*d^7*e^8 - 3*b*c^2*d^6*e^9 + 3*b^2*c*d^5*e^10 -
b^3*d^4*e^11)*x^3 + 6*(c^3*d^8*e^7 - 3*b*c^2*d^7*e^8 + 3*b^2*c*d^6*e^9 - b^3*d^5*e^10)*x^2 + 4*(c^3*d^9*e^6 -
3*b*c^2*d^8*e^7 + 3*b^2*c*d^7*e^8 - b^3*d^6*e^9)*x), -1/192*(3*(128*B*c^4*d^9 - 320*B*b*c^3*d^8*e + 240*B*b^2
*c^2*d^7*e^2 - 40*B*b^3*c*d^6*e^3 - 5*B*b^4*d^5*e^4 - 3*A*b^4*d^4*e^5 + (128*B*c^4*d^5*e^4 - 320*B*b*c^3*d^4*e
^5 + 240*B*b^2*c^2*d^3*e^6 - 40*B*b^3*c*d^2*e^7 - 5*B*b^4*d*e^8 - 3*A*b^4*e^9)*x^4 + 4*(128*B*c^4*d^6*e^3 - 32
0*B*b*c^3*d^5*e^4 + 240*B*b^2*c^2*d^4*e^5 - 40*B*b^3*c*d^3*e^6 - 5*B*b^4*d^2*e^7 - 3*A*b^4*d*e^8)*x^3 + 6*(128
*B*c^4*d^7*e^2 - 320*B*b*c^3*d^6*e^3 + 240*B*b^2*c^2*d^5*e^4 - 40*B*b^3*c*d^4*e^5 - 5*B*b^4*d^3*e^6 - 3*A*b^4*
d^2*e^7)*x^2 + 4*(128*B*c^4*d^8*e - 320*B*b*c^3*d^7*e^2 + 240*B*b^2*c^2*d^6*e^3 - 40*B*b^3*c*d^5*e^4 - 5*B*b^4
*d^4*e^5 - 3*A*b^4*d^3*e^6)*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)) - 192*(B*c^4*d^10 - 3*B*b*c^3*d^9*e + 3*B*b^2*c^2*d^8*e^2 - B*b^3*c*d^7*e^3 + (B*c^4*d^6*e^4 - 3*B*b*c^3
*d^5*e^5 + 3*B*b^2*c^2*d^4*e^6 - B*b^3*c*d^3*e^7)*x^4 + 4*(B*c^4*d^7*e^3 - 3*B*b*c^3*d^6*e^4 + 3*B*b^2*c^2*d^5
*e^5 - B*b^3*c*d^4*e^6)*x^3 + 6*(B*c^4*d^8*e^2 - 3*B*b*c^3*d^7*e^3 + 3*B*b^2*c^2*d^6*e^4 - B*b^3*c*d^5*e^5)*x^
2 + 4*(B*c^4*d^9*e - 3*B*b*c^3*d^8*e^2 + 3*B*b^2*c^2*d^7*e^3 - B*b^3*c*d^6*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*s
qrt(c*x^2 + b*x)*sqrt(c)) + (192*B*c^4*d^9*e - 528*B*b*c^3*d^8*e^2 + 456*B*b^2*c^2*d^7*e^3 - 105*B*b^3*c*d^6*e
^4 - 9*A*b^4*d^4*e^6 - 3*(5*B*b^4 - 3*A*b^3*c)*d^5*e^5 + (400*B*c^4*d^6*e^4 + 9*A*b^4*d*e^9 - 24*(49*B*b*c^3 +
2*A*c^4)*d^5*e^5 + 6*(193*B*b^2*c^2 + 20*A*b*c^3)*d^4*e^6 - (397*B*b^3*c + 78*A*b^2*c^2)*d^3*e^7 + 3*(5*B*b^4
- A*b^3*c)*d^2*e^8)*x^3 + (832*B*c^4*d^7*e^3 - 2312*B*b*c^3*d^6*e^4 + 33*A*b^4*d^2*e^8 + 12*(169*B*b^2*c^2 -
6*A*b*c^3)*d^5*e^5 - (475*B*b^3*c - 204*A*b^2*c^2)*d^4*e^6 - (73*B*b^4 + 165*A*b^3*c)*d^3*e^7)*x^2 + (672*B*c^
4*d^8*e^2 - 1856*B*b*c^3*d^7*e^3 + 1614*B*b^2*c^2*d^6*e^4 - 33*A*b^4*d^3*e^7 - 3*(125*B*b^3*c + 2*A*b^2*c^2)*d
^5*e^5 - (55*B*b^4 - 39*A*b^3*c)*d^4*e^6)*x)*sqrt(c*x^2 + b*x))/(c^3*d^10*e^5 - 3*b*c^2*d^9*e^6 + 3*b^2*c*d^8*
e^7 - b^3*d^7*e^8 + (c^3*d^6*e^9 - 3*b*c^2*d^5*e^10 + 3*b^2*c*d^4*e^11 - b^3*d^3*e^12)*x^4 + 4*(c^3*d^7*e^8 -
3*b*c^2*d^6*e^9 + 3*b^2*c*d^5*e^10 - b^3*d^4*e^11)*x^3 + 6*(c^3*d^8*e^7 - 3*b*c^2*d^7*e^8 + 3*b^2*c*d^6*e^9 -
b^3*d^5*e^10)*x^2 + 4*(c^3*d^9*e^6 - 3*b*c^2*d^8*e^7 + 3*b^2*c*d^7*e^8 - b^3*d^6*e^9)*x), -1/384*(768*(B*c^4*d
^10 - 3*B*b*c^3*d^9*e + 3*B*b^2*c^2*d^8*e^2 - B*b^3*c*d^7*e^3 + (B*c^4*d^6*e^4 - 3*B*b*c^3*d^5*e^5 + 3*B*b^2*c
^2*d^4*e^6 - B*b^3*c*d^3*e^7)*x^4 + 4*(B*c^4*d^7*e^3 - 3*B*b*c^3*d^6*e^4 + 3*B*b^2*c^2*d^5*e^5 - B*b^3*c*d^4*e
^6)*x^3 + 6*(B*c^4*d^8*e^2 - 3*B*b*c^3*d^7*e^3 + 3*B*b^2*c^2*d^6*e^4 - B*b^3*c*d^5*e^5)*x^2 + 4*(B*c^4*d^9*e -
3*B*b*c^3*d^8*e^2 + 3*B*b^2*c^2*d^7*e^3 - B*b^3*c*d^6*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x
)) + 3*(128*B*c^4*d^9 - 320*B*b*c^3*d^8*e + 240*B*b^2*c^2*d^7*e^2 - 40*B*b^3*c*d^6*e^3 - 5*B*b^4*d^5*e^4 - 3*A
*b^4*d^4*e^5 + (128*B*c^4*d^5*e^4 - 320*B*b*c^3*d^4*e^5 + 240*B*b^2*c^2*d^3*e^6 - 40*B*b^3*c*d^2*e^7 - 5*B*b^4
*d*e^8 - 3*A*b^4*e^9)*x^4 + 4*(128*B*c^4*d^6*e^3 - 320*B*b*c^3*d^5*e^4 + 240*B*b^2*c^2*d^4*e^5 - 40*B*b^3*c*d^
3*e^6 - 5*B*b^4*d^2*e^7 - 3*A*b^4*d*e^8)*x^3 + 6*(128*B*c^4*d^7*e^2 - 320*B*b*c^3*d^6*e^3 + 240*B*b^2*c^2*d^5*
e^4 - 40*B*b^3*c*d^4*e^5 - 5*B*b^4*d^3*e^6 - 3*A*b^4*d^2*e^7)*x^2 + 4*(128*B*c^4*d^8*e - 320*B*b*c^3*d^7*e^2 +
240*B*b^2*c^2*d^6*e^3 - 40*B*b^3*c*d^5*e^4 - 5*B*b^4*d^4*e^5 - 3*A*b^4*d^3*e^6)*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*(192*B*c^4*d^9*e - 528*B*b*c^3*
d^8*e^2 + 456*B*b^2*c^2*d^7*e^3 - 105*B*b^3*c*d^6*e^4 - 9*A*b^4*d^4*e^6 - 3*(5*B*b^4 - 3*A*b^3*c)*d^5*e^5 + (4
00*B*c^4*d^6*e^4 + 9*A*b^4*d*e^9 - 24*(49*B*b*c^3 + 2*A*c^4)*d^5*e^5 + 6*(193*B*b^2*c^2 + 20*A*b*c^3)*d^4*e^6
- (397*B*b^3*c + 78*A*b^2*c^2)*d^3*e^7 + 3*(5*B*b^4 - A*b^3*c)*d^2*e^8)*x^3 + (832*B*c^4*d^7*e^3 - 2312*B*b*c^
3*d^6*e^4 + 33*A*b^4*d^2*e^8 + 12*(169*B*b^2*c^2 - 6*A*b*c^3)*d^5*e^5 - (475*B*b^3*c - 204*A*b^2*c^2)*d^4*e^6
- (73*B*b^4 + 165*A*b^3*c)*d^3*e^7)*x^2 + (672*B*c^4*d^8*e^2 - 1856*B*b*c^3*d^7*e^3 + 1614*B*b^2*c^2*d^6*e^4 -
33*A*b^4*d^3*e^7 - 3*(125*B*b^3*c + 2*A*b^2*c^2)*d^5*e^5 - (55*B*b^4 - 39*A*b^3*c)*d^4*e^6)*x)*sqrt(c*x^2 + b
*x))/(c^3*d^10*e^5 - 3*b*c^2*d^9*e^6 + 3*b^2*c*d^8*e^7 - b^3*d^7*e^8 + (c^3*d^6*e^9 - 3*b*c^2*d^5*e^10 + 3*b^2
*c*d^4*e^11 - b^3*d^3*e^12)*x^4 + 4*(c^3*d^7*e^8 - 3*b*c^2*d^6*e^9 + 3*b^2*c*d^5*e^10 - b^3*d^4*e^11)*x^3 + 6*
(c^3*d^8*e^7 - 3*b*c^2*d^7*e^8 + 3*b^2*c*d^6*e^9 - b^3*d^5*e^10)*x^2 + 4*(c^3*d^9*e^6 - 3*b*c^2*d^8*e^7 + 3*b^
2*c*d^7*e^8 - b^3*d^6*e^9)*x), -1/192*(3*(128*B*c^4*d^9 - 320*B*b*c^3*d^8*e + 240*B*b^2*c^2*d^7*e^2 - 40*B*b^3
*c*d^6*e^3 - 5*B*b^4*d^5*e^4 - 3*A*b^4*d^4*e^5 + (128*B*c^4*d^5*e^4 - 320*B*b*c^3*d^4*e^5 + 240*B*b^2*c^2*d^3*
e^6 - 40*B*b^3*c*d^2*e^7 - 5*B*b^4*d*e^8 - 3*A*b^4*e^9)*x^4 + 4*(128*B*c^4*d^6*e^3 - 320*B*b*c^3*d^5*e^4 + 240
*B*b^2*c^2*d^4*e^5 - 40*B*b^3*c*d^3*e^6 - 5*B*b^4*d^2*e^7 - 3*A*b^4*d*e^8)*x^3 + 6*(128*B*c^4*d^7*e^2 - 320*B*
b*c^3*d^6*e^3 + 240*B*b^2*c^2*d^5*e^4 - 40*B*b^3*c*d^4*e^5 - 5*B*b^4*d^3*e^6 - 3*A*b^4*d^2*e^7)*x^2 + 4*(128*B
*c^4*d^8*e - 320*B*b*c^3*d^7*e^2 + 240*B*b^2*c^2*d^6*e^3 - 40*B*b^3*c*d^5*e^4 - 5*B*b^4*d^4*e^5 - 3*A*b^4*d^3*
e^6)*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)) + 384*(B*c^4*d^10
- 3*B*b*c^3*d^9*e + 3*B*b^2*c^2*d^8*e^2 - B*b^3*c*d^7*e^3 + (B*c^4*d^6*e^4 - 3*B*b*c^3*d^5*e^5 + 3*B*b^2*c^2*
d^4*e^6 - B*b^3*c*d^3*e^7)*x^4 + 4*(B*c^4*d^7*e^3 - 3*B*b*c^3*d^6*e^4 + 3*B*b^2*c^2*d^5*e^5 - B*b^3*c*d^4*e^6)
*x^3 + 6*(B*c^4*d^8*e^2 - 3*B*b*c^3*d^7*e^3 + 3*B*b^2*c^2*d^6*e^4 - B*b^3*c*d^5*e^5)*x^2 + 4*(B*c^4*d^9*e - 3*
B*b*c^3*d^8*e^2 + 3*B*b^2*c^2*d^7*e^3 - B*b^3*c*d^6*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x))
+ (192*B*c^4*d^9*e - 528*B*b*c^3*d^8*e^2 + 456*B*b^2*c^2*d^7*e^3 - 105*B*b^3*c*d^6*e^4 - 9*A*b^4*d^4*e^6 - 3*(
5*B*b^4 - 3*A*b^3*c)*d^5*e^5 + (400*B*c^4*d^6*e^4 + 9*A*b^4*d*e^9 - 24*(49*B*b*c^3 + 2*A*c^4)*d^5*e^5 + 6*(193
*B*b^2*c^2 + 20*A*b*c^3)*d^4*e^6 - (397*B*b^3*c + 78*A*b^2*c^2)*d^3*e^7 + 3*(5*B*b^4 - A*b^3*c)*d^2*e^8)*x^3 +
(832*B*c^4*d^7*e^3 - 2312*B*b*c^3*d^6*e^4 + 33*A*b^4*d^2*e^8 + 12*(169*B*b^2*c^2 - 6*A*b*c^3)*d^5*e^5 - (475*
B*b^3*c - 204*A*b^2*c^2)*d^4*e^6 - (73*B*b^4 + 165*A*b^3*c)*d^3*e^7)*x^2 + (672*B*c^4*d^8*e^2 - 1856*B*b*c^3*d
^7*e^3 + 1614*B*b^2*c^2*d^6*e^4 - 33*A*b^4*d^3*e^7 - 3*(125*B*b^3*c + 2*A*b^2*c^2)*d^5*e^5 - (55*B*b^4 - 39*A*
b^3*c)*d^4*e^6)*x)*sqrt(c*x^2 + b*x))/(c^3*d^10*e^5 - 3*b*c^2*d^9*e^6 + 3*b^2*c*d^8*e^7 - b^3*d^7*e^8 + (c^3*d
^6*e^9 - 3*b*c^2*d^5*e^10 + 3*b^2*c*d^4*e^11 - b^3*d^3*e^12)*x^4 + 4*(c^3*d^7*e^8 - 3*b*c^2*d^6*e^9 + 3*b^2*c*
d^5*e^10 - b^3*d^4*e^11)*x^3 + 6*(c^3*d^8*e^7 - 3*b*c^2*d^7*e^8 + 3*b^2*c*d^6*e^9 - b^3*d^5*e^10)*x^2 + 4*(c^3
*d^9*e^6 - 3*b*c^2*d^8*e^7 + 3*b^2*c*d^7*e^8 - b^3*d^6*e^9)*x)]

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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)**5,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)^5,x, algorithm="giac")

[Out]

Exception raised: TypeError