3.1189 \(\int \frac{(A+B x) (b x+c x^2)^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=495 \[ -\frac{5 \sqrt{b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{8 e^6 (d+e x)}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (2 A c e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (24 b^2 c d e^2-b^3 e^3-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 \sqrt{c} e^7}+\frac{5 \left (B d \left (56 b^2 c d e^2-7 b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (18 b^2 c d e^2-b^3 e^3-48 b c^2 d^2 e+32 c^3 d^3\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 \sqrt{d} e^7 \sqrt{c d-b e}}+\frac{\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac{5 \left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{12 e^4 (d+e x)^2} \]

[Out]

(-5*(A*e*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2) - 2*B*d*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) + e*(4*A*c*e*(2*c*d
- b*e) - B*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(8*e^6*(d + e*x)) - (5*(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)*(b*x + c*x^2)^(3/2))/(12*e^4*(d + e*x)^2) + ((2*B*
d - A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(3*e^2*(d + e*x)^3) + (5*(2*A*c*e*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)
- B*(64*c^3*d^3 - 80*b*c^2*d^2*e + 24*b^2*c*d*e^2 - b^3*e^3))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*Sqrt[
c]*e^7) + (5*(B*d*(64*c^3*d^3 - 112*b*c^2*d^2*e + 56*b^2*c*d*e^2 - 7*b^3*e^3) - A*e*(32*c^3*d^3 - 48*b*c^2*d^2
*e + 18*b^2*c*d*e^2 - b^3*e^3))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])]
)/(16*Sqrt[d]*e^7*Sqrt[c*d - b*e])

________________________________________________________________________________________

Rubi [A]  time = 0.662105, antiderivative size = 495, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{5 \sqrt{b x+c x^2} \left (e x \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )+A e \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )-2 B d \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )}{8 e^6 (d+e x)}+\frac{5 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (2 A c e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B \left (24 b^2 c d e^2-b^3 e^3-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 \sqrt{c} e^7}+\frac{5 \left (B d \left (56 b^2 c d e^2-7 b^3 e^3-112 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (18 b^2 c d e^2-b^3 e^3-48 b c^2 d^2 e+32 c^3 d^3\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 \sqrt{d} e^7 \sqrt{c d-b e}}+\frac{\left (b x+c x^2\right )^{5/2} (-A e+2 B d+B e x)}{3 e^2 (d+e x)^3}-\frac{5 \left (b x+c x^2\right )^{3/2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{12 e^4 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(-5*(A*e*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2) - 2*B*d*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) + e*(4*A*c*e*(2*c*d
- b*e) - B*(16*c^2*d^2 - 12*b*c*d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(8*e^6*(d + e*x)) - (5*(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)*(b*x + c*x^2)^(3/2))/(12*e^4*(d + e*x)^2) + ((2*B*
d - A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(3*e^2*(d + e*x)^3) + (5*(2*A*c*e*(16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)
- B*(64*c^3*d^3 - 80*b*c^2*d^2*e + 24*b^2*c*d*e^2 - b^3*e^3))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*Sqrt[
c]*e^7) + (5*(B*d*(64*c^3*d^3 - 112*b*c^2*d^2*e + 56*b^2*c*d*e^2 - 7*b^3*e^3) - A*e*(32*c^3*d^3 - 48*b*c^2*d^2
*e + 18*b^2*c*d*e^2 - b^3*e^3))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])]
)/(16*Sqrt[d]*e^7*Sqrt[c*d - b*e])

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]

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 )^{5/2}}{(d+e x)^4} \, dx &=\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac{5 \int \frac{(3 b (2 B d-A e)+3 (4 B c d-b B e-2 A c e) x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx}{18 e^2}\\ &=-\frac{5 (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) \left (b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac{5 \int \frac{\left (6 b (4 B d (2 c d-b e)-A e (4 c d-b e))-6 \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\right ) \sqrt{b x+c x^2}}{(d+e x)^2} \, dx}{48 e^4}\\ &=-\frac{5 \left (A e \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )-2 B d \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )+e \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\right ) \sqrt{b x+c x^2}}{8 e^6 (d+e x)}-\frac{5 (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) \left (b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac{5 \int \frac{-6 b \left (A e \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )-2 B d \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )\right )-6 \left (2 A c e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+24 b^2 c d e^2-b^3 e^3\right )\right ) x}{(d+e x) \sqrt{b x+c x^2}} \, dx}{96 e^6}\\ &=-\frac{5 \left (A e \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )-2 B d \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )+e \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\right ) \sqrt{b x+c x^2}}{8 e^6 (d+e x)}-\frac{5 (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) \left (b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac{\left (5 \left (B d \left (64 c^3 d^3-112 b c^2 d^2 e+56 b^2 c d e^2-7 b^3 e^3\right )-A e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{16 e^7}+\frac{\left (5 \left (2 A c e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+24 b^2 c d e^2-b^3 e^3\right )\right )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{16 e^7}\\ &=-\frac{5 \left (A e \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )-2 B d \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )+e \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\right ) \sqrt{b x+c x^2}}{8 e^6 (d+e x)}-\frac{5 (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) \left (b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}-\frac{\left (5 \left (B d \left (64 c^3 d^3-112 b c^2 d^2 e+56 b^2 c d e^2-7 b^3 e^3\right )-A e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\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 )}{8 e^7}+\frac{\left (5 \left (2 A c e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+24 b^2 c d e^2-b^3 e^3\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{8 e^7}\\ &=-\frac{5 \left (A e \left (16 c^2 d^2-12 b c d e+b^2 e^2\right )-2 B d \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )+e \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\right ) \sqrt{b x+c x^2}}{8 e^6 (d+e x)}-\frac{5 (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) \left (b x+c x^2\right )^{3/2}}{12 e^4 (d+e x)^2}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{5/2}}{3 e^2 (d+e x)^3}+\frac{5 \left (2 A c e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+24 b^2 c d e^2-b^3 e^3\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 \sqrt{c} e^7}+\frac{5 \left (B d \left (64 c^3 d^3-112 b c^2 d^2 e+56 b^2 c d e^2-7 b^3 e^3\right )-A e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\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 )}{16 \sqrt{d} e^7 \sqrt{c d-b e}}\\ \end{align*}

Mathematica [B]  time = 6.25797, size = 2249, normalized size = 4.54 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(5/2))/(3*d*(-(c*d) + b*e)*(d + e*x)^3) + ((x*(b + c*x))^(5/2)*(((-4
*c*d*(B*d - A*e) + (e*(7*b*B*d - 6*A*c*d - A*b*e))/2)*x^(7/2)*(b + c*x)^(7/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2
) + ((((e*(24*A*c^2*d^2 + 2*b*c*d*(14*B*d - 17*A*e) - 3*b^2*e*(7*B*d - A*e)))/4 - (5*c*d*(B*d*(8*c*d - 7*b*e)
- A*e*(2*c*d - b*e)))/2)*x^(7/2)*(b + c*x)^(7/2))/(d*(-(c*d) + b*e)*(d + e*x)) + ((-3*(16*A*c^3*d^3 + 2*b^2*c*
d*e*(98*B*d - 39*A*e) - 8*b*c^2*d^2*(21*B*d - 8*A*e) - 5*b^3*e^2*(7*B*d - A*e))*((2*b^2*x^(5/2)*Sqrt[b + c*x]*
(1 + (c*x)/b)^3*((5/(16*(1 + (c*x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/2 - (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])))/(
512*c^3*x^3*(1 + (c*x)/b)^3)))/(5*e) - (d*((2*b^2*x^(3/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((3*(5/(8*(1 + (c*x)/b
)^3) + 5/(6*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/8 + (15*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])))/(256*c^2*x^2*(1 + (c*x)/b)^3)))/(3*e) - (d*((2*b^2*Sqrt[x]
*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((15/(8*(1 + (c*x)/b)^3) + 5/(4*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1))/6 + (5*S
qrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(16*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(7/2))))/e - (d*((2*b*c*Sqrt[x]*S
qrt[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]*Sqr
t[x])/Sqrt[b]])/(8*Sqrt[c]*Sqrt[x]*(1 + (c*x)/b)^(5/2))))/e - ((c*d - b*e)*((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))/e))/8 - (3*c*(A*e*(44*c^2*d^2 - 44*b*c*d*e + 3*b^2*e^2) - B*d*(80*c^2*d^2 - 98*b*c*d*e
+ 21*b^2*e^2))*((2*b^2*x^(7/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((7*(3/(16*(1 + (c*x)/b)^3) + 1/(2*(1 + (c*x)/b)
^2) + (1 + (c*x)/b)^(-1)))/12 + (35*b^4*((2*c*x)/b - (4*c^2*x^2)/(3*b^2) + (16*c^3*x^3)/(15*b^3) - (2*Sqrt[c]*
Sqrt[x]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(2048*c^4*x^4*(1 + (c*x)/b)^3)))/(7*
e) - (d*((2*b^2*x^(5/2)*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((5/(16*(1 + (c*x)/b)^3) + 5/(8*(1 + (c*x)/b)^2) + (1 +
(c*x)/b)^(-1))/2 - (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])/Sqr
t[b]])/(Sqrt[b]*Sqrt[1 + (c*x)/b])))/(512*c^3*x^3*(1 + (c*x)/b)^3)))/(5*e) - (d*((2*b^2*x^(3/2)*Sqrt[b + c*x]*
(1 + (c*x)/b)^3*((3*(5/(8*(1 + (c*x)/b)^3) + 5/(6*(1 + (c*x)/b)^2) + (1 + (c*x)/b)^(-1)))/8 + (15*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])))/(256*c^2*x^2*(1 + (c
*x)/b)^3)))/(3*e) - (d*((2*b^2*Sqrt[x]*Sqrt[b + c*x]*(1 + (c*x)/b)^3*((15/(8*(1 + (c*x)/b)^3) + 5/(4*(1 + (c*x
)/b)^2) + (1 + (c*x)/b)^(-1))/6 + (5*Sqrt[b]*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]])/(16*Sqrt[c]*Sqrt[x]*(1 + (c*x
)/b)^(7/2))))/e - (d*((2*b*c*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 - ((c*d - b*
e)*((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[(S
qrt[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))/e))/e))/2)/(d*(-(c*d) + b*e)))/(2*d*(-(c*d) + b*
e))))/(3*d*(-(c*d) + b*e)*x^(5/2)*(b + c*x)^(5/2))

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Maple [B]  time = 0.022, size = 17133, normalized size = 34.6 \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)^(5/2)/(e*x+d)^4,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)^(5/2)/(e*x+d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 24.1801, size = 12497, normalized size = 25.25 \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)^(5/2)/(e*x+d)^4,x, algorithm="fricas")

[Out]

[-1/48*(15*(64*B*c^4*d^8 - 16*(9*B*b*c^3 + 2*A*c^4)*d^7*e + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^2 - (25*B*b^3*c
+ 38*A*b^2*c^2)*d^5*e^3 + (B*b^4 + 6*A*b^3*c)*d^4*e^4 + (64*B*c^4*d^5*e^3 - 16*(9*B*b*c^3 + 2*A*c^4)*d^4*e^4
+ 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^5 - (25*B*b^3*c + 38*A*b^2*c^2)*d^2*e^6 + (B*b^4 + 6*A*b^3*c)*d*e^7)*x^3
+ 3*(64*B*c^4*d^6*e^2 - 16*(9*B*b*c^3 + 2*A*c^4)*d^5*e^3 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^4*e^4 - (25*B*b^3*c
+ 38*A*b^2*c^2)*d^3*e^5 + (B*b^4 + 6*A*b^3*c)*d^2*e^6)*x^2 + 3*(64*B*c^4*d^7*e - 16*(9*B*b*c^3 + 2*A*c^4)*d^6*
e^2 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^3 - (25*B*b^3*c + 38*A*b^2*c^2)*d^4*e^4 + (B*b^4 + 6*A*b^3*c)*d^3*e^5
)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 15*(64*B*c^4*d^7 + A*b^3*c*d^3*e^4 - 16*(7*B*b*c^3
+ 2*A*c^4)*d^6*e + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^5*e^2 - (7*B*b^3*c + 18*A*b^2*c^2)*d^4*e^3 + (64*B*c^4*d^4*e
^3 + A*b^3*c*e^7 - 16*(7*B*b*c^3 + 2*A*c^4)*d^3*e^4 + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^2*e^5 - (7*B*b^3*c + 18*A*
b^2*c^2)*d*e^6)*x^3 + 3*(64*B*c^4*d^5*e^2 + A*b^3*c*d*e^6 - 16*(7*B*b*c^3 + 2*A*c^4)*d^4*e^3 + 8*(7*B*b^2*c^2
+ 6*A*b*c^3)*d^3*e^4 - (7*B*b^3*c + 18*A*b^2*c^2)*d^2*e^5)*x^2 + 3*(64*B*c^4*d^6*e + A*b^3*c*d^2*e^5 - 16*(7*B
*b*c^3 + 2*A*c^4)*d^5*e^2 + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^4*e^3 - (7*B*b^3*c + 18*A*b^2*c^2)*d^3*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*(480*B*c^4
*d^7*e + 15*A*b^3*c*d^3*e^5 - 240*(4*B*b*c^3 + A*c^4)*d^6*e^2 + 30*(19*B*b^2*c^2 + 14*A*b*c^3)*d^5*e^3 - 15*(6
*B*b^3*c + 13*A*b^2*c^2)*d^4*e^4 + 8*(B*c^4*d^2*e^6 - B*b*c^3*d*e^7)*x^5 - 2*(12*B*c^4*d^3*e^5 - (25*B*b*c^3 +
6*A*c^4)*d^2*e^6 + (13*B*b^2*c^2 + 6*A*b*c^3)*d*e^7)*x^4 + 3*(40*B*c^4*d^4*e^4 - 2*(43*B*b*c^3 + 10*A*c^4)*d^
3*e^5 + 19*(3*B*b^2*c^2 + 2*A*b*c^3)*d^2*e^6 - (11*B*b^3*c + 18*A*b^2*c^2)*d*e^7)*x^3 + (880*B*c^4*d^5*e^3 + 3
3*A*b^3*c*d*e^7 - 40*(45*B*b*c^3 + 11*A*c^4)*d^4*e^4 + 158*(7*B*b^2*c^2 + 5*A*b*c^3)*d^3*e^5 - (186*B*b^3*c +
383*A*b^2*c^2)*d^2*e^6)*x^2 + 5*(240*B*c^4*d^6*e^2 + 8*A*b^3*c*d^2*e^6 - 4*(121*B*b*c^3 + 30*A*c^4)*d^5*e^3 +
(291*B*b^2*c^2 + 212*A*b*c^3)*d^4*e^4 - (47*B*b^3*c + 100*A*b^2*c^2)*d^3*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*d^5*e
^7 - b*c*d^4*e^8 + (c^2*d^2*e^10 - b*c*d*e^11)*x^3 + 3*(c^2*d^3*e^9 - b*c*d^2*e^10)*x^2 + 3*(c^2*d^4*e^8 - b*c
*d^3*e^9)*x), 1/48*(30*(64*B*c^4*d^7 + A*b^3*c*d^3*e^4 - 16*(7*B*b*c^3 + 2*A*c^4)*d^6*e + 8*(7*B*b^2*c^2 + 6*A
*b*c^3)*d^5*e^2 - (7*B*b^3*c + 18*A*b^2*c^2)*d^4*e^3 + (64*B*c^4*d^4*e^3 + A*b^3*c*e^7 - 16*(7*B*b*c^3 + 2*A*c
^4)*d^3*e^4 + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^2*e^5 - (7*B*b^3*c + 18*A*b^2*c^2)*d*e^6)*x^3 + 3*(64*B*c^4*d^5*e^
2 + A*b^3*c*d*e^6 - 16*(7*B*b*c^3 + 2*A*c^4)*d^4*e^3 + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^3*e^4 - (7*B*b^3*c + 18*A
*b^2*c^2)*d^2*e^5)*x^2 + 3*(64*B*c^4*d^6*e + A*b^3*c*d^2*e^5 - 16*(7*B*b*c^3 + 2*A*c^4)*d^5*e^2 + 8*(7*B*b^2*c
^2 + 6*A*b*c^3)*d^4*e^3 - (7*B*b^3*c + 18*A*b^2*c^2)*d^3*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)) - 15*(64*B*c^4*d^8 - 16*(9*B*b*c^3 + 2*A*c^4)*d^7*e + 8*(13*B*b^2*c^2
+ 8*A*b*c^3)*d^6*e^2 - (25*B*b^3*c + 38*A*b^2*c^2)*d^5*e^3 + (B*b^4 + 6*A*b^3*c)*d^4*e^4 + (64*B*c^4*d^5*e^3 -
16*(9*B*b*c^3 + 2*A*c^4)*d^4*e^4 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^5 - (25*B*b^3*c + 38*A*b^2*c^2)*d^2*e^6
+ (B*b^4 + 6*A*b^3*c)*d*e^7)*x^3 + 3*(64*B*c^4*d^6*e^2 - 16*(9*B*b*c^3 + 2*A*c^4)*d^5*e^3 + 8*(13*B*b^2*c^2 +
8*A*b*c^3)*d^4*e^4 - (25*B*b^3*c + 38*A*b^2*c^2)*d^3*e^5 + (B*b^4 + 6*A*b^3*c)*d^2*e^6)*x^2 + 3*(64*B*c^4*d^7
*e - 16*(9*B*b*c^3 + 2*A*c^4)*d^6*e^2 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^3 - (25*B*b^3*c + 38*A*b^2*c^2)*d^4
*e^4 + (B*b^4 + 6*A*b^3*c)*d^3*e^5)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(480*B*c^4*d^7
*e + 15*A*b^3*c*d^3*e^5 - 240*(4*B*b*c^3 + A*c^4)*d^6*e^2 + 30*(19*B*b^2*c^2 + 14*A*b*c^3)*d^5*e^3 - 15*(6*B*b
^3*c + 13*A*b^2*c^2)*d^4*e^4 + 8*(B*c^4*d^2*e^6 - B*b*c^3*d*e^7)*x^5 - 2*(12*B*c^4*d^3*e^5 - (25*B*b*c^3 + 6*A
*c^4)*d^2*e^6 + (13*B*b^2*c^2 + 6*A*b*c^3)*d*e^7)*x^4 + 3*(40*B*c^4*d^4*e^4 - 2*(43*B*b*c^3 + 10*A*c^4)*d^3*e^
5 + 19*(3*B*b^2*c^2 + 2*A*b*c^3)*d^2*e^6 - (11*B*b^3*c + 18*A*b^2*c^2)*d*e^7)*x^3 + (880*B*c^4*d^5*e^3 + 33*A*
b^3*c*d*e^7 - 40*(45*B*b*c^3 + 11*A*c^4)*d^4*e^4 + 158*(7*B*b^2*c^2 + 5*A*b*c^3)*d^3*e^5 - (186*B*b^3*c + 383*
A*b^2*c^2)*d^2*e^6)*x^2 + 5*(240*B*c^4*d^6*e^2 + 8*A*b^3*c*d^2*e^6 - 4*(121*B*b*c^3 + 30*A*c^4)*d^5*e^3 + (291
*B*b^2*c^2 + 212*A*b*c^3)*d^4*e^4 - (47*B*b^3*c + 100*A*b^2*c^2)*d^3*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*d^5*e^7 -
b*c*d^4*e^8 + (c^2*d^2*e^10 - b*c*d*e^11)*x^3 + 3*(c^2*d^3*e^9 - b*c*d^2*e^10)*x^2 + 3*(c^2*d^4*e^8 - b*c*d^3
*e^9)*x), 1/48*(30*(64*B*c^4*d^8 - 16*(9*B*b*c^3 + 2*A*c^4)*d^7*e + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^6*e^2 - (25
*B*b^3*c + 38*A*b^2*c^2)*d^5*e^3 + (B*b^4 + 6*A*b^3*c)*d^4*e^4 + (64*B*c^4*d^5*e^3 - 16*(9*B*b*c^3 + 2*A*c^4)*
d^4*e^4 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^5 - (25*B*b^3*c + 38*A*b^2*c^2)*d^2*e^6 + (B*b^4 + 6*A*b^3*c)*d*e
^7)*x^3 + 3*(64*B*c^4*d^6*e^2 - 16*(9*B*b*c^3 + 2*A*c^4)*d^5*e^3 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^4*e^4 - (25*
B*b^3*c + 38*A*b^2*c^2)*d^3*e^5 + (B*b^4 + 6*A*b^3*c)*d^2*e^6)*x^2 + 3*(64*B*c^4*d^7*e - 16*(9*B*b*c^3 + 2*A*c
^4)*d^6*e^2 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^3 - (25*B*b^3*c + 38*A*b^2*c^2)*d^4*e^4 + (B*b^4 + 6*A*b^3*c)
*d^3*e^5)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - 15*(64*B*c^4*d^7 + A*b^3*c*d^3*e^4 - 16*(7*B*
b*c^3 + 2*A*c^4)*d^6*e + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^5*e^2 - (7*B*b^3*c + 18*A*b^2*c^2)*d^4*e^3 + (64*B*c^4*
d^4*e^3 + A*b^3*c*e^7 - 16*(7*B*b*c^3 + 2*A*c^4)*d^3*e^4 + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^2*e^5 - (7*B*b^3*c +
18*A*b^2*c^2)*d*e^6)*x^3 + 3*(64*B*c^4*d^5*e^2 + A*b^3*c*d*e^6 - 16*(7*B*b*c^3 + 2*A*c^4)*d^4*e^3 + 8*(7*B*b^2
*c^2 + 6*A*b*c^3)*d^3*e^4 - (7*B*b^3*c + 18*A*b^2*c^2)*d^2*e^5)*x^2 + 3*(64*B*c^4*d^6*e + A*b^3*c*d^2*e^5 - 16
*(7*B*b*c^3 + 2*A*c^4)*d^5*e^2 + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^4*e^3 - (7*B*b^3*c + 18*A*b^2*c^2)*d^3*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*(480*
B*c^4*d^7*e + 15*A*b^3*c*d^3*e^5 - 240*(4*B*b*c^3 + A*c^4)*d^6*e^2 + 30*(19*B*b^2*c^2 + 14*A*b*c^3)*d^5*e^3 -
15*(6*B*b^3*c + 13*A*b^2*c^2)*d^4*e^4 + 8*(B*c^4*d^2*e^6 - B*b*c^3*d*e^7)*x^5 - 2*(12*B*c^4*d^3*e^5 - (25*B*b*
c^3 + 6*A*c^4)*d^2*e^6 + (13*B*b^2*c^2 + 6*A*b*c^3)*d*e^7)*x^4 + 3*(40*B*c^4*d^4*e^4 - 2*(43*B*b*c^3 + 10*A*c^
4)*d^3*e^5 + 19*(3*B*b^2*c^2 + 2*A*b*c^3)*d^2*e^6 - (11*B*b^3*c + 18*A*b^2*c^2)*d*e^7)*x^3 + (880*B*c^4*d^5*e^
3 + 33*A*b^3*c*d*e^7 - 40*(45*B*b*c^3 + 11*A*c^4)*d^4*e^4 + 158*(7*B*b^2*c^2 + 5*A*b*c^3)*d^3*e^5 - (186*B*b^3
*c + 383*A*b^2*c^2)*d^2*e^6)*x^2 + 5*(240*B*c^4*d^6*e^2 + 8*A*b^3*c*d^2*e^6 - 4*(121*B*b*c^3 + 30*A*c^4)*d^5*e
^3 + (291*B*b^2*c^2 + 212*A*b*c^3)*d^4*e^4 - (47*B*b^3*c + 100*A*b^2*c^2)*d^3*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*
d^5*e^7 - b*c*d^4*e^8 + (c^2*d^2*e^10 - b*c*d*e^11)*x^3 + 3*(c^2*d^3*e^9 - b*c*d^2*e^10)*x^2 + 3*(c^2*d^4*e^8
- b*c*d^3*e^9)*x), 1/24*(15*(64*B*c^4*d^7 + A*b^3*c*d^3*e^4 - 16*(7*B*b*c^3 + 2*A*c^4)*d^6*e + 8*(7*B*b^2*c^2
+ 6*A*b*c^3)*d^5*e^2 - (7*B*b^3*c + 18*A*b^2*c^2)*d^4*e^3 + (64*B*c^4*d^4*e^3 + A*b^3*c*e^7 - 16*(7*B*b*c^3 +
2*A*c^4)*d^3*e^4 + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^2*e^5 - (7*B*b^3*c + 18*A*b^2*c^2)*d*e^6)*x^3 + 3*(64*B*c^4*d
^5*e^2 + A*b^3*c*d*e^6 - 16*(7*B*b*c^3 + 2*A*c^4)*d^4*e^3 + 8*(7*B*b^2*c^2 + 6*A*b*c^3)*d^3*e^4 - (7*B*b^3*c +
18*A*b^2*c^2)*d^2*e^5)*x^2 + 3*(64*B*c^4*d^6*e + A*b^3*c*d^2*e^5 - 16*(7*B*b*c^3 + 2*A*c^4)*d^5*e^2 + 8*(7*B*
b^2*c^2 + 6*A*b*c^3)*d^4*e^3 - (7*B*b^3*c + 18*A*b^2*c^2)*d^3*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)) + 15*(64*B*c^4*d^8 - 16*(9*B*b*c^3 + 2*A*c^4)*d^7*e + 8*(13*B*b^2
*c^2 + 8*A*b*c^3)*d^6*e^2 - (25*B*b^3*c + 38*A*b^2*c^2)*d^5*e^3 + (B*b^4 + 6*A*b^3*c)*d^4*e^4 + (64*B*c^4*d^5*
e^3 - 16*(9*B*b*c^3 + 2*A*c^4)*d^4*e^4 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^3*e^5 - (25*B*b^3*c + 38*A*b^2*c^2)*d^
2*e^6 + (B*b^4 + 6*A*b^3*c)*d*e^7)*x^3 + 3*(64*B*c^4*d^6*e^2 - 16*(9*B*b*c^3 + 2*A*c^4)*d^5*e^3 + 8*(13*B*b^2*
c^2 + 8*A*b*c^3)*d^4*e^4 - (25*B*b^3*c + 38*A*b^2*c^2)*d^3*e^5 + (B*b^4 + 6*A*b^3*c)*d^2*e^6)*x^2 + 3*(64*B*c^
4*d^7*e - 16*(9*B*b*c^3 + 2*A*c^4)*d^6*e^2 + 8*(13*B*b^2*c^2 + 8*A*b*c^3)*d^5*e^3 - (25*B*b^3*c + 38*A*b^2*c^2
)*d^4*e^4 + (B*b^4 + 6*A*b^3*c)*d^3*e^5)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (480*B*c^4*d^7
*e + 15*A*b^3*c*d^3*e^5 - 240*(4*B*b*c^3 + A*c^4)*d^6*e^2 + 30*(19*B*b^2*c^2 + 14*A*b*c^3)*d^5*e^3 - 15*(6*B*b
^3*c + 13*A*b^2*c^2)*d^4*e^4 + 8*(B*c^4*d^2*e^6 - B*b*c^3*d*e^7)*x^5 - 2*(12*B*c^4*d^3*e^5 - (25*B*b*c^3 + 6*A
*c^4)*d^2*e^6 + (13*B*b^2*c^2 + 6*A*b*c^3)*d*e^7)*x^4 + 3*(40*B*c^4*d^4*e^4 - 2*(43*B*b*c^3 + 10*A*c^4)*d^3*e^
5 + 19*(3*B*b^2*c^2 + 2*A*b*c^3)*d^2*e^6 - (11*B*b^3*c + 18*A*b^2*c^2)*d*e^7)*x^3 + (880*B*c^4*d^5*e^3 + 33*A*
b^3*c*d*e^7 - 40*(45*B*b*c^3 + 11*A*c^4)*d^4*e^4 + 158*(7*B*b^2*c^2 + 5*A*b*c^3)*d^3*e^5 - (186*B*b^3*c + 383*
A*b^2*c^2)*d^2*e^6)*x^2 + 5*(240*B*c^4*d^6*e^2 + 8*A*b^3*c*d^2*e^6 - 4*(121*B*b*c^3 + 30*A*c^4)*d^5*e^3 + (291
*B*b^2*c^2 + 212*A*b*c^3)*d^4*e^4 - (47*B*b^3*c + 100*A*b^2*c^2)*d^3*e^5)*x)*sqrt(c*x^2 + b*x))/(c^2*d^5*e^7 -
b*c*d^4*e^8 + (c^2*d^2*e^10 - b*c*d*e^11)*x^3 + 3*(c^2*d^3*e^9 - b*c*d^2*e^10)*x^2 + 3*(c^2*d^4*e^8 - b*c*d^3
*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)**(5/2)/(e*x+d)**4,x)

[Out]

Timed out

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Giac [B]  time = 2.3253, size = 2574, normalized size = 5.2 \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)^(5/2)/(e*x+d)^4,x, algorithm="giac")

[Out]

5/8*(64*B*c^3*d^4 - 112*B*b*c^2*d^3*e - 32*A*c^3*d^3*e + 56*B*b^2*c*d^2*e^2 + 48*A*b*c^2*d^2*e^2 - 7*B*b^3*d*e
^3 - 18*A*b^2*c*d*e^3 + A*b^3*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e
))*e^(-7)/sqrt(-c*d^2 + b*d*e) + 5/16*(64*B*c^3*d^3 - 80*B*b*c^2*d^2*e - 32*A*c^3*d^2*e + 24*B*b^2*c*d*e^2 + 3
2*A*b*c^2*d*e^2 - B*b^3*e^3 - 6*A*b^2*c*e^3)*e^(-7)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/sq
rt(c) + 1/24*sqrt(c*x^2 + b*x)*(2*(4*B*c^2*x*e^(-4) - (24*B*c^4*d*e^17 - 13*B*b*c^3*e^18 - 6*A*c^4*e^18)*e^(-2
2)/c^2)*x + 3*(80*B*c^4*d^2*e^16 - 72*B*b*c^3*d*e^17 - 32*A*c^4*d*e^17 + 11*B*b^2*c^2*e^18 + 18*A*b*c^3*e^18)*
e^(-22)/c^2) + 1/24*(2592*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*c^4*d^5*e + 2368*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^3*B*c^(9/2)*d^6 + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*c^(7/2)*d^4*e^2 - 1168*(sqrt(c)*x - sqrt(c*x^2 + b*
x))^3*B*b*c^(7/2)*d^5*e - 1504*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(9/2)*d^5*e + 3552*(sqrt(c)*x - sqrt(c*x^
2 + b*x))^2*B*b*c^4*d^6 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b*c^3*d^4*e^2 - 1680*(sqrt(c)*x - sqrt(c*x^
2 + b*x))^4*A*c^4*d^4*e^2 - 4512*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^2*c^3*d^5*e - 2256*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^2*A*b*c^4*d^5*e + 1776*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^2*c^(7/2)*d^6 - 1200*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^5*B*b*c^(5/2)*d^3*e^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^(7/2)*d^3*e^3 - 1920*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^3*B*b^2*c^(5/2)*d^4*e^2 + 400*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^(7/2)*d^4*e^2 - 2
340*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*c^(5/2)*d^5*e - 1128*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^2*c^(7/2)*d
^5*e + 296*B*b^3*c^3*d^6 + 1560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c^2*d^3*e^3 + 2160*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^4*A*b*c^3*d^3*e^3 + 1314*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*c^2*d^4*e^2 + 2412*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^2*A*b^2*c^3*d^4*e^2 - 350*B*b^4*c^2*d^5*e - 188*A*b^3*c^3*d^5*e + 600*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^5*B*b^2*c^(3/2)*d^2*e^4 + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b*c^(5/2)*d^2*e^4 + 1186*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^3*B*b^3*c^(3/2)*d^3*e^3 + 1308*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^(5/2)*d^3*e^3
+ 786*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*c^(3/2)*d^4*e^2 + 1272*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*c^(5
/2)*d^4*e^2 - 147*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*c*d^2*e^4 - 666*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*
b^2*c^2*d^2*e^4 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4*c*d^3*e^3 - 462*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2
*A*b^3*c^2*d^3*e^3 + 87*B*b^5*c*d^4*e^2 + 188*A*b^4*c^2*d^4*e^2 - 87*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*s
qrt(c)*d*e^5 - 306*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^(3/2)*d*e^5 - 136*(sqrt(c)*x - sqrt(c*x^2 + b*x))
^3*B*b^4*sqrt(c)*d^2*e^4 - 574*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c^(3/2)*d^2*e^4 - 57*(sqrt(c)*x - sqrt(
c*x^2 + b*x))*B*b^5*sqrt(c)*d^3*e^3 - 324*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*c^(3/2)*d^3*e^3 + 21*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^4*A*b^3*c*d*e^5 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^4*c*d^2*e^4 - 33*A*b^5*c*d^3
*e^3 + 33*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^3*sqrt(c)*e^6 + 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*sqr
t(c)*d*e^5 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*sqrt(c)*d^2*e^4)*e^(-7)/(((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)^3*sqrt(c))