3.1177 \(\int \frac{(A+B x) (b x+c x^2)^{3/2}}{(d+e x)^3} \, dx\)
Optimal. Leaf size=314 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{4 \sqrt{c} e^5}+\frac{3 \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 \sqrt{d} e^5 \sqrt{c d-b e}}+\frac{\left (b x+c x^2\right )^{3/2} (-A e+2 B d+B e x)}{2 e^2 (d+e x)^2}-\frac{3 \sqrt{b x+c x^2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{4 e^4 (d+e x)} \]
[Out]
(-3*(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)*Sqrt[b*x + c*x^2])/(4*e^4*(d +
e*x)) + ((2*B*d - A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(2*e^2*(d + e*x)^2) - (3*(4*A*c*e*(2*c*d - b*e) - B*(16*c
^2*d^2 - 12*b*c*d*e + b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*Sqrt[c]*e^5) + (3*(A*e*(8*c^2*d^2 -
8*b*c*d*e + b^2*e^2) - B*d*(16*c^2*d^2 - 20*b*c*d*e + 5*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*Sqrt[d]*e^5*Sqrt[c*d - b*e])
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Rubi [A] time = 0.426819, antiderivative size = 314, 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 =
{812, 843, 620, 206, 724} \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (b^2 e^2-12 b c d e+16 c^2 d^2\right )\right )}{4 \sqrt{c} e^5}+\frac{3 \left (A e \left (b^2 e^2-8 b c d e+8 c^2 d^2\right )-B d \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 \sqrt{d} e^5 \sqrt{c d-b e}}+\frac{\left (b x+c x^2\right )^{3/2} (-A e+2 B d+B e x)}{2 e^2 (d+e x)^2}-\frac{3 \sqrt{b x+c x^2} (e x (-2 A c e-b B e+4 B c d)-A e (4 c d-b e)+4 B d (2 c d-b e))}{4 e^4 (d+e x)} \]
Antiderivative was successfully verified.
[In]
Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^3,x]
[Out]
(-3*(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)*Sqrt[b*x + c*x^2])/(4*e^4*(d +
e*x)) + ((2*B*d - A*e + B*e*x)*(b*x + c*x^2)^(3/2))/(2*e^2*(d + e*x)^2) - (3*(4*A*c*e*(2*c*d - b*e) - B*(16*c
^2*d^2 - 12*b*c*d*e + b^2*e^2))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(4*Sqrt[c]*e^5) + (3*(A*e*(8*c^2*d^2 -
8*b*c*d*e + b^2*e^2) - B*d*(16*c^2*d^2 - 20*b*c*d*e + 5*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*Sqrt[d]*e^5*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 )^{3/2}}{(d+e x)^3} \, dx &=\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac{3 \int \frac{(2 b (2 B d-A e)+2 (4 B c d-b B e-2 A c e) x) \sqrt{b x+c x^2}}{(d+e x)^2} \, dx}{8 e^2}\\ &=-\frac{3 (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) \sqrt{b x+c x^2}}{4 e^4 (d+e x)}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}+\frac{3 \int \frac{2 b (4 B d (2 c d-b e)-A e (4 c d-b e))-2 \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}{(d+e x) \sqrt{b x+c x^2}} \, dx}{16 e^4}\\ &=-\frac{3 (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) \sqrt{b x+c x^2}}{4 e^4 (d+e x)}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac{\left (3 \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 )\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{8 e^5}+\frac{\left (3 \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{8 e^5}\\ &=-\frac{3 (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) \sqrt{b x+c x^2}}{4 e^4 (d+e x)}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac{\left (3 \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 )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{4 e^5}-\frac{\left (3 \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\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 )}{4 e^5}\\ &=-\frac{3 (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) \sqrt{b x+c x^2}}{4 e^4 (d+e x)}+\frac{(2 B d-A e+B e x) \left (b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}-\frac{3 \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 ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 \sqrt{c} e^5}+\frac{3 \left (A e \left (8 c^2 d^2-8 b c d e+b^2 e^2\right )-B d \left (16 c^2 d^2-20 b c d e+5 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 \sqrt{d} e^5 \sqrt{c d-b e}}\\ \end{align*}
Mathematica [B] time = 6.12435, size = 1374, normalized size = 4.38 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^3,x]
[Out]
((-(B*d) + A*e)*x*(b + c*x)*(x*(b + c*x))^(3/2))/(2*d*(-(c*d) + b*e)*(d + e*x)^2) + ((x*(b + c*x))^(3/2)*(((-3
*c*d*(B*d - A*e) + (e*(5*b*B*d - 4*A*c*d - A*b*e))/2)*x^(5/2)*(b + c*x)^(5/2))/(d*(-(c*d) + b*e)*(d + e*x)) +
(((8*A*c^2*d^2 + 4*b*c*d*(5*B*d - 4*A*e) - 3*b^2*e*(5*B*d - A*e))*((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])/Sq
rt[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)
)/4 + 2*c*(B*d*(6*c*d - 5*b*e) - A*e*(2*c*d - b*e))*((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])/Sqrt[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*Sq
rt[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*S
qrt[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*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)*x^(3/2)*(b + c*x)^(3/2))
________________________________________________________________________________________
Maple [B] time = 0.013, size = 7365, normalized size = 23.5 \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)^3,x)
[Out]
result too large to display
________________________________________________________________________________________
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)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 10.2156, size = 6975, normalized size = 22.21 \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)^3,x, algorithm="fricas")
[Out]
[1/8*(3*(16*B*c^3*d^6 - 4*(7*B*b*c^2 + 2*A*c^3)*d^5*e + (13*B*b^2*c + 12*A*b*c^2)*d^4*e^2 - (B*b^3 + 4*A*b^2*c
)*d^3*e^3 + (16*B*c^3*d^4*e^2 - 4*(7*B*b*c^2 + 2*A*c^3)*d^3*e^3 + (13*B*b^2*c + 12*A*b*c^2)*d^2*e^4 - (B*b^3 +
4*A*b^2*c)*d*e^5)*x^2 + 2*(16*B*c^3*d^5*e - 4*(7*B*b*c^2 + 2*A*c^3)*d^4*e^2 + (13*B*b^2*c + 12*A*b*c^2)*d^3*e
^3 - (B*b^3 + 4*A*b^2*c)*d^2*e^4)*x)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 3*(16*B*c^3*d^5 -
A*b^2*c*d^2*e^3 - 4*(5*B*b*c^2 + 2*A*c^3)*d^4*e + (5*B*b^2*c + 8*A*b*c^2)*d^3*e^2 + (16*B*c^3*d^3*e^2 - A*b^2*
c*e^5 - 4*(5*B*b*c^2 + 2*A*c^3)*d^2*e^3 + (5*B*b^2*c + 8*A*b*c^2)*d*e^4)*x^2 + 2*(16*B*c^3*d^4*e - A*b^2*c*d*e
^4 - 4*(5*B*b*c^2 + 2*A*c^3)*d^3*e^2 + (5*B*b^2*c + 8*A*b*c^2)*d^2*e^3)*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*(24*B*c^3*d^5*e - 3*A*b^2*c*d^2*e^4 - 12
*(3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(4*B*b^2*c + 5*A*b*c^2)*d^3*e^3 - 2*(B*c^3*d^2*e^4 - B*b*c^2*d*e^5)*x^3 + (8*
B*c^3*d^3*e^3 - (13*B*b*c^2 + 4*A*c^3)*d^2*e^4 + (5*B*b^2*c + 4*A*b*c^2)*d*e^5)*x^2 + (36*B*c^3*d^4*e^2 - 5*A*
b^2*c*d*e^5 - (55*B*b*c^2 + 18*A*c^3)*d^3*e^3 + (19*B*b^2*c + 23*A*b*c^2)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*
d^4*e^5 - b*c*d^3*e^6 + (c^2*d^2*e^7 - b*c*d*e^8)*x^2 + 2*(c^2*d^3*e^6 - b*c*d^2*e^7)*x), -1/8*(6*(16*B*c^3*d^
5 - A*b^2*c*d^2*e^3 - 4*(5*B*b*c^2 + 2*A*c^3)*d^4*e + (5*B*b^2*c + 8*A*b*c^2)*d^3*e^2 + (16*B*c^3*d^3*e^2 - A*
b^2*c*e^5 - 4*(5*B*b*c^2 + 2*A*c^3)*d^2*e^3 + (5*B*b^2*c + 8*A*b*c^2)*d*e^4)*x^2 + 2*(16*B*c^3*d^4*e - A*b^2*c
*d*e^4 - 4*(5*B*b*c^2 + 2*A*c^3)*d^3*e^2 + (5*B*b^2*c + 8*A*b*c^2)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(-sq
rt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - 3*(16*B*c^3*d^6 - 4*(7*B*b*c^2 + 2*A*c^3)*d^5*e + (13*
B*b^2*c + 12*A*b*c^2)*d^4*e^2 - (B*b^3 + 4*A*b^2*c)*d^3*e^3 + (16*B*c^3*d^4*e^2 - 4*(7*B*b*c^2 + 2*A*c^3)*d^3*
e^3 + (13*B*b^2*c + 12*A*b*c^2)*d^2*e^4 - (B*b^3 + 4*A*b^2*c)*d*e^5)*x^2 + 2*(16*B*c^3*d^5*e - 4*(7*B*b*c^2 +
2*A*c^3)*d^4*e^2 + (13*B*b^2*c + 12*A*b*c^2)*d^3*e^3 - (B*b^3 + 4*A*b^2*c)*d^2*e^4)*x)*sqrt(c)*log(2*c*x + b +
2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(24*B*c^3*d^5*e - 3*A*b^2*c*d^2*e^4 - 12*(3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(4*
B*b^2*c + 5*A*b*c^2)*d^3*e^3 - 2*(B*c^3*d^2*e^4 - B*b*c^2*d*e^5)*x^3 + (8*B*c^3*d^3*e^3 - (13*B*b*c^2 + 4*A*c^
3)*d^2*e^4 + (5*B*b^2*c + 4*A*b*c^2)*d*e^5)*x^2 + (36*B*c^3*d^4*e^2 - 5*A*b^2*c*d*e^5 - (55*B*b*c^2 + 18*A*c^3
)*d^3*e^3 + (19*B*b^2*c + 23*A*b*c^2)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^4*e^5 - b*c*d^3*e^6 + (c^2*d^2*e^7
- b*c*d*e^8)*x^2 + 2*(c^2*d^3*e^6 - b*c*d^2*e^7)*x), -1/8*(6*(16*B*c^3*d^6 - 4*(7*B*b*c^2 + 2*A*c^3)*d^5*e +
(13*B*b^2*c + 12*A*b*c^2)*d^4*e^2 - (B*b^3 + 4*A*b^2*c)*d^3*e^3 + (16*B*c^3*d^4*e^2 - 4*(7*B*b*c^2 + 2*A*c^3)*
d^3*e^3 + (13*B*b^2*c + 12*A*b*c^2)*d^2*e^4 - (B*b^3 + 4*A*b^2*c)*d*e^5)*x^2 + 2*(16*B*c^3*d^5*e - 4*(7*B*b*c^
2 + 2*A*c^3)*d^4*e^2 + (13*B*b^2*c + 12*A*b*c^2)*d^3*e^3 - (B*b^3 + 4*A*b^2*c)*d^2*e^4)*x)*sqrt(-c)*arctan(sqr
t(c*x^2 + b*x)*sqrt(-c)/(c*x)) - 3*(16*B*c^3*d^5 - A*b^2*c*d^2*e^3 - 4*(5*B*b*c^2 + 2*A*c^3)*d^4*e + (5*B*b^2*
c + 8*A*b*c^2)*d^3*e^2 + (16*B*c^3*d^3*e^2 - A*b^2*c*e^5 - 4*(5*B*b*c^2 + 2*A*c^3)*d^2*e^3 + (5*B*b^2*c + 8*A*
b*c^2)*d*e^4)*x^2 + 2*(16*B*c^3*d^4*e - A*b^2*c*d*e^4 - 4*(5*B*b*c^2 + 2*A*c^3)*d^3*e^2 + (5*B*b^2*c + 8*A*b*c
^2)*d^2*e^3)*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*(24*B*c^3*d^5*e - 3*A*b^2*c*d^2*e^4 - 12*(3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(4*B*b^2*c + 5*A*b*c^2)*d^
3*e^3 - 2*(B*c^3*d^2*e^4 - B*b*c^2*d*e^5)*x^3 + (8*B*c^3*d^3*e^3 - (13*B*b*c^2 + 4*A*c^3)*d^2*e^4 + (5*B*b^2*c
+ 4*A*b*c^2)*d*e^5)*x^2 + (36*B*c^3*d^4*e^2 - 5*A*b^2*c*d*e^5 - (55*B*b*c^2 + 18*A*c^3)*d^3*e^3 + (19*B*b^2*c
+ 23*A*b*c^2)*d^2*e^4)*x)*sqrt(c*x^2 + b*x))/(c^2*d^4*e^5 - b*c*d^3*e^6 + (c^2*d^2*e^7 - b*c*d*e^8)*x^2 + 2*(
c^2*d^3*e^6 - b*c*d^2*e^7)*x), -1/4*(3*(16*B*c^3*d^5 - A*b^2*c*d^2*e^3 - 4*(5*B*b*c^2 + 2*A*c^3)*d^4*e + (5*B*
b^2*c + 8*A*b*c^2)*d^3*e^2 + (16*B*c^3*d^3*e^2 - A*b^2*c*e^5 - 4*(5*B*b*c^2 + 2*A*c^3)*d^2*e^3 + (5*B*b^2*c +
8*A*b*c^2)*d*e^4)*x^2 + 2*(16*B*c^3*d^4*e - A*b^2*c*d*e^4 - 4*(5*B*b*c^2 + 2*A*c^3)*d^3*e^2 + (5*B*b^2*c + 8*A
*b*c^2)*d^2*e^3)*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)) + 3*(
16*B*c^3*d^6 - 4*(7*B*b*c^2 + 2*A*c^3)*d^5*e + (13*B*b^2*c + 12*A*b*c^2)*d^4*e^2 - (B*b^3 + 4*A*b^2*c)*d^3*e^3
+ (16*B*c^3*d^4*e^2 - 4*(7*B*b*c^2 + 2*A*c^3)*d^3*e^3 + (13*B*b^2*c + 12*A*b*c^2)*d^2*e^4 - (B*b^3 + 4*A*b^2*
c)*d*e^5)*x^2 + 2*(16*B*c^3*d^5*e - 4*(7*B*b*c^2 + 2*A*c^3)*d^4*e^2 + (13*B*b^2*c + 12*A*b*c^2)*d^3*e^3 - (B*b
^3 + 4*A*b^2*c)*d^2*e^4)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (24*B*c^3*d^5*e - 3*A*b^2*c*d^
2*e^4 - 12*(3*B*b*c^2 + A*c^3)*d^4*e^2 + 3*(4*B*b^2*c + 5*A*b*c^2)*d^3*e^3 - 2*(B*c^3*d^2*e^4 - B*b*c^2*d*e^5)
*x^3 + (8*B*c^3*d^3*e^3 - (13*B*b*c^2 + 4*A*c^3)*d^2*e^4 + (5*B*b^2*c + 4*A*b*c^2)*d*e^5)*x^2 + (36*B*c^3*d^4*
e^2 - 5*A*b^2*c*d*e^5 - (55*B*b*c^2 + 18*A*c^3)*d^3*e^3 + (19*B*b^2*c + 23*A*b*c^2)*d^2*e^4)*x)*sqrt(c*x^2 + b
*x))/(c^2*d^4*e^5 - b*c*d^3*e^6 + (c^2*d^2*e^7 - b*c*d*e^8)*x^2 + 2*(c^2*d^3*e^6 - b*c*d^2*e^7)*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)**3,x)
[Out]
Timed out
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Giac [B] time = 1.70055, size = 1264, normalized size = 4.03 \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)^3,x, algorithm="giac")
[Out]
-3/4*(16*B*c^2*d^3 - 20*B*b*c*d^2*e - 8*A*c^2*d^2*e + 5*B*b^2*d*e^2 + 8*A*b*c*d*e^2 - A*b^2*e^3)*arctan(-((sqr
t(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))*e^(-5)/sqrt(-c*d^2 + b*d*e) - 3/8*(16*B*c^2*d
^2 - 12*B*b*c*d*e - 8*A*c^2*d*e + B*b^2*e^2 + 4*A*b*c*e^2)*e^(-5)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sq
rt(c) + b))/sqrt(c) + 1/4*(2*B*c*x*e^(-3) - (12*B*c^2*d*e^8 - 5*B*b*c*e^9 - 4*A*c^2*e^9)*e^(-12)/c)*sqrt(c*x^2
+ b*x) - 1/4*(32*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^(5/2)*d^3*e + 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*c
^3*d^4 - 44*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b*c^2*d^3*e - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*c^3*d^3*e
+ 56*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b*c^(5/2)*d^4 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c^(3/2)*d^2*e
^2 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*c^(5/2)*d^2*e^2 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^2*c^(3/2)
*d^3*e - 40*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b*c^(5/2)*d^3*e + 14*B*b^2*c^2*d^4 + 3*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^2*B*b^2*c*d^2*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b*c^2*d^2*e^2 - 9*B*b^3*c*d^3*e - 10*A*b^2*c
^2*d^3*e + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*sqrt(c)*d*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*
c^(3/2)*d*e^3 + 7*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^3*sqrt(c)*d^2*e^2 + 28*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A
*b^2*c^(3/2)*d^2*e^2 + (sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^2*c*d*e^3 + 5*A*b^3*c*d^2*e^2 - 5*(sqrt(c)*x - sq
rt(c*x^2 + b*x))^3*A*b^2*sqrt(c)*e^4 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^3*sqrt(c)*d*e^3)*e^(-5)/(((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*sqrt(c))