3.1171 \(\int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^6} \, dx\)

Optimal. Leaf size=449 \[ \frac{\left (b x+c x^2\right )^{3/2} \left (B d \left (-15 b^2 e^2+42 b c d e+8 c^2 d^2\right )-A e \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}+\frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (6 b^2 c d e (5 A e+2 B d)+b^3 \left (-e^2\right ) (7 A e+3 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right )}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{b^2 \left (6 b^2 c d e (5 A e+2 B d)+b^3 \left (-e^2\right ) (7 A e+3 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}-\frac{\left (b x+c x^2\right )^{3/2} (7 A e (2 c d-b e)-B d (3 b e+4 c d))}{40 d^2 (d+e x)^4 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{3/2} (B d-A e)}{5 d (d+e x)^5 (c d-b e)} \]

[Out]

((32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e) + 6*b^2*c*d*e*(2*B*d + 5*A*e) - b^3*e^2*(3*B*d + 7*A*e))*(b*d + (2
*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^4*(c*d - b*e)^4*(d + e*x)^2) + ((B*d - A*e)*(b*x + c*x^2)^(3/2))/(5*d
*(c*d - b*e)*(d + e*x)^5) - ((7*A*e*(2*c*d - b*e) - B*d*(4*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(40*d^2*(c*d - b
*e)^2*(d + e*x)^4) + ((B*d*(8*c^2*d^2 + 42*b*c*d*e - 15*b^2*e^2) - A*e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2
))*(b*x + c*x^2)^(3/2))/(240*d^3*(c*d - b*e)^3*(d + e*x)^3) - (b^2*(32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e)
+ 6*b^2*c*d*e*(2*B*d + 5*A*e) - b^3*e^2*(3*B*d + 7*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d -
b*e]*Sqrt[b*x + c*x^2])])/(256*d^(9/2)*(c*d - b*e)^(9/2))

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Rubi [A]  time = 0.781627, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {834, 806, 720, 724, 206} \[ \frac{\left (b x+c x^2\right )^{3/2} \left (B d \left (-15 b^2 e^2+42 b c d e+8 c^2 d^2\right )-A e \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )\right )}{240 d^3 (d+e x)^3 (c d-b e)^3}+\frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d) \left (6 b^2 c d e (5 A e+2 B d)+b^3 \left (-e^2\right ) (7 A e+3 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right )}{128 d^4 (d+e x)^2 (c d-b e)^4}-\frac{b^2 \left (6 b^2 c d e (5 A e+2 B d)+b^3 \left (-e^2\right ) (7 A e+3 B d)-16 b c^2 d^2 (3 A e+B d)+32 A c^3 d^3\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{256 d^{9/2} (c d-b e)^{9/2}}-\frac{\left (b x+c x^2\right )^{3/2} (7 A e (2 c d-b e)-B d (3 b e+4 c d))}{40 d^2 (d+e x)^4 (c d-b e)^2}+\frac{\left (b x+c x^2\right )^{3/2} (B d-A e)}{5 d (d+e x)^5 (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^6,x]

[Out]

((32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e) + 6*b^2*c*d*e*(2*B*d + 5*A*e) - b^3*e^2*(3*B*d + 7*A*e))*(b*d + (2
*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(128*d^4*(c*d - b*e)^4*(d + e*x)^2) + ((B*d - A*e)*(b*x + c*x^2)^(3/2))/(5*d
*(c*d - b*e)*(d + e*x)^5) - ((7*A*e*(2*c*d - b*e) - B*d*(4*c*d + 3*b*e))*(b*x + c*x^2)^(3/2))/(40*d^2*(c*d - b
*e)^2*(d + e*x)^4) + ((B*d*(8*c^2*d^2 + 42*b*c*d*e - 15*b^2*e^2) - A*e*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2
))*(b*x + c*x^2)^(3/2))/(240*d^3*(c*d - b*e)^3*(d + e*x)^3) - (b^2*(32*A*c^3*d^3 - 16*b*c^2*d^2*(B*d + 3*A*e)
+ 6*b^2*c*d*e*(2*B*d + 5*A*e) - b^3*e^2*(3*B*d + 7*A*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d -
b*e]*Sqrt[b*x + c*x^2])])/(256*d^(9/2)*(c*d - b*e)^(9/2))

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*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] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(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] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 720

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*
(d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^p)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(p*(b^2 -
4*a*c))/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[
{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m +
2*p + 2, 0] && GtQ[p, 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]

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])

Rubi steps

\begin{align*} \int \frac{(A+B x) \sqrt{b x+c x^2}}{(d+e x)^6} \, dx &=\frac{(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{\int \frac{\left (\frac{1}{2} (-10 A c d+b (3 B d+7 A e))-2 c (B d-A e) x\right ) \sqrt{b x+c x^2}}{(d+e x)^5} \, dx}{5 d (c d-b e)}\\ &=\frac{(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac{\int \frac{\left (\frac{1}{4} \left (80 A c^2 d^2+5 b^2 e (3 B d+7 A e)-2 b c d (18 B d+47 A e)\right )-\frac{1}{2} c (7 A e (2 c d-b e)-B d (4 c d+3 b e)) x\right ) \sqrt{b x+c x^2}}{(d+e x)^4} \, dx}{20 d^2 (c d-b e)^2}\\ &=\frac{(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac{\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}+\frac{\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) \int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx}{32 d^3 (c d-b e)^3}\\ &=\frac{\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}+\frac{(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac{\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac{\left (b^2 \left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right )\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{256 d^4 (c d-b e)^4}\\ &=\frac{\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}+\frac{(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac{\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}+\frac{\left (b^2 \left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\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 )}{128 d^4 (c d-b e)^4}\\ &=\frac{\left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt{b x+c x^2}}{128 d^4 (c d-b e)^4 (d+e x)^2}+\frac{(B d-A e) \left (b x+c x^2\right )^{3/2}}{5 d (c d-b e) (d+e x)^5}-\frac{(7 A e (2 c d-b e)-B d (4 c d+3 b e)) \left (b x+c x^2\right )^{3/2}}{40 d^2 (c d-b e)^2 (d+e x)^4}+\frac{\left (B d \left (8 c^2 d^2+42 b c d e-15 b^2 e^2\right )-A e \left (108 c^2 d^2-108 b c d e+35 b^2 e^2\right )\right ) \left (b x+c x^2\right )^{3/2}}{240 d^3 (c d-b e)^3 (d+e x)^3}-\frac{b^2 \left (32 A c^3 d^3-16 b c^2 d^2 (B d+3 A e)+6 b^2 c d e (2 B d+5 A e)-b^3 e^2 (3 B d+7 A e)\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 )}{256 d^{9/2} (c d-b e)^{9/2}}\\ \end{align*}

Mathematica [A]  time = 2.31938, size = 387, normalized size = 0.86 \[ \frac{\sqrt{x (b+c x)} \left (\frac{(d+e x)^2 \left (8 x^{3/2} (b+c x) \left (A e \left (35 b^2 e^2-108 b c d e+108 c^2 d^2\right )+B d \left (15 b^2 e^2-42 b c d e-8 c^2 d^2\right )\right )+\frac{15 (d+e x) \left (-6 b^2 c d e (5 A e+2 B d)+b^3 e^2 (7 A e+3 B d)+16 b c^2 d^2 (3 A e+B d)-32 A c^3 d^3\right ) \left (b^2 (d+e x)^2 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )+\sqrt{d} \sqrt{x} \sqrt{b+c x} \sqrt{b e-c d} (-b d+b e x-2 c d x)\right )}{d^{3/2} \sqrt{b+c x} (b e-c d)^{3/2}}\right )}{d^2 (c d-b e)^2}-\frac{48 x^{3/2} (b+c x) (d+e x) (7 A e (b e-2 c d)+B d (3 b e+4 c d))}{d (c d-b e)}+384 x^{3/2} (b+c x) (A e-B d)\right )}{1920 d \sqrt{x} (d+e x)^5 (b e-c d)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x)^6,x]

[Out]

(Sqrt[x*(b + c*x)]*(384*(-(B*d) + A*e)*x^(3/2)*(b + c*x) - (48*(7*A*e*(-2*c*d + b*e) + B*d*(4*c*d + 3*b*e))*x^
(3/2)*(b + c*x)*(d + e*x))/(d*(c*d - b*e)) + ((d + e*x)^2*(8*(B*d*(-8*c^2*d^2 - 42*b*c*d*e + 15*b^2*e^2) + A*e
*(108*c^2*d^2 - 108*b*c*d*e + 35*b^2*e^2))*x^(3/2)*(b + c*x) + (15*(-32*A*c^3*d^3 + 16*b*c^2*d^2*(B*d + 3*A*e)
- 6*b^2*c*d*e*(2*B*d + 5*A*e) + b^3*e^2*(3*B*d + 7*A*e))*(d + e*x)*(Sqrt[d]*Sqrt[-(c*d) + b*e]*Sqrt[x]*Sqrt[b
+ c*x]*(-(b*d) - 2*c*d*x + b*e*x) + b^2*(d + e*x)^2*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x
])]))/(d^(3/2)*(-(c*d) + b*e)^(3/2)*Sqrt[b + c*x])))/(d^2*(c*d - b*e)^2)))/(1920*d*(-(c*d) + b*e)*Sqrt[x]*(d +
e*x)^5)

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Maple [B]  time = 0.021, size = 15015, normalized size = 33.4 \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)^(1/2)/(e*x+d)^6,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)^(1/2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.38349, size = 7301, normalized size = 16.26 \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)^(1/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

[1/3840*(15*(7*A*b^5*d^5*e^3 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^8 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^7*e + 3*(B*b^5
- 10*A*b^4*c)*d^6*e^2 + (7*A*b^5*e^8 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3*e^5 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^2*e
^6 + 3*(B*b^5 - 10*A*b^4*c)*d*e^7)*x^5 + 5*(7*A*b^5*d*e^7 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4*e^4 - 12*(B*b^4*c
- 4*A*b^3*c^2)*d^3*e^5 + 3*(B*b^5 - 10*A*b^4*c)*d^2*e^6)*x^4 + 10*(7*A*b^5*d^2*e^6 + 16*(B*b^3*c^2 - 2*A*b^2*
c^3)*d^5*e^3 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^4*e^4 + 3*(B*b^5 - 10*A*b^4*c)*d^3*e^5)*x^3 + 10*(7*A*b^5*d^3*e^5
+ 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^6*e^2 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^5*e^3 + 3*(B*b^5 - 10*A*b^4*c)*d^4*e^4)*
x^2 + 5*(7*A*b^5*d^4*e^4 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^7*e - 12*(B*b^4*c - 4*A*b^3*c^2)*d^6*e^2 + 3*(B*b^5
- 10*A*b^4*c)*d^5*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*(105*A*b^5*d^5*e^4 - 240*(B*b^2*c^3 - 2*A*b*c^4)*d^9 + 60*(7*B*b^3*c^2 - 20*A*b^2*c^3)*d^
8*e - 45*(5*B*b^4*c - 26*A*b^3*c^2)*d^7*e^2 + 15*(3*B*b^5 - 37*A*b^4*c)*d^6*e^3 + (64*B*c^5*d^7*e^2 - 105*A*b^
5*d*e^8 - 16*(17*B*b*c^4 - 6*A*c^5)*d^6*e^3 + 32*(11*B*b^2*c^3 - 9*A*b*c^4)*d^5*e^4 - 2*(147*B*b^3*c^2 - 334*A
*b^2*c^3)*d^4*e^5 + (195*B*b^4*c - 856*A*b^3*c^2)*d^3*e^6 - 5*(9*B*b^5 - 97*A*b^4*c)*d^2*e^7)*x^4 + 2*(160*B*c
^5*d^8*e - 245*A*b^5*d^2*e^7 - 24*(29*B*b*c^4 - 10*A*c^5)*d^7*e^2 + 12*(79*B*b^2*c^3 - 62*A*b*c^4)*d^6*e^3 - (
763*B*b^3*c^2 - 1622*A*b^2*c^3)*d^5*e^4 + 3*(152*B*b^4*c - 669*A*b^3*c^2)*d^4*e^5 - 21*(5*B*b^5 - 54*A*b^4*c)*
d^3*e^6)*x^3 + 2*(320*B*c^5*d^9 - 448*A*b^5*d^3*e^6 - 480*(3*B*b*c^4 - A*c^5)*d^8*e + 24*(88*B*b^2*c^3 - 65*A*
b*c^4)*d^7*e^2 - (1691*B*b^3*c^2 - 3178*A*b^2*c^3)*d^6*e^3 + 33*(27*B*b^4*c - 113*A*b^3*c^2)*d^5*e^4 - 3*(64*B
*b^5 - 693*A*b^4*c)*d^4*e^5)*x^2 - 10*(79*A*b^5*d^4*e^5 - 16*(B*b*c^4 + 6*A*c^5)*d^9 + 28*(5*B*b^2*c^3 + 12*A*
b*c^4)*d^8*e - (211*B*b^3*c^2 + 642*A*b^2*c^3)*d^7*e^2 + (108*B*b^4*c + 697*A*b^3*c^2)*d^6*e^3 - (21*B*b^5 + 3
74*A*b^4*c)*d^5*e^4)*x)*sqrt(c*x^2 + b*x))/(c^5*d^15 - 5*b*c^4*d^14*e + 10*b^2*c^3*d^13*e^2 - 10*b^3*c^2*d^12*
e^3 + 5*b^4*c*d^11*e^4 - b^5*d^10*e^5 + (c^5*d^10*e^5 - 5*b*c^4*d^9*e^6 + 10*b^2*c^3*d^8*e^7 - 10*b^3*c^2*d^7*
e^8 + 5*b^4*c*d^6*e^9 - b^5*d^5*e^10)*x^5 + 5*(c^5*d^11*e^4 - 5*b*c^4*d^10*e^5 + 10*b^2*c^3*d^9*e^6 - 10*b^3*c
^2*d^8*e^7 + 5*b^4*c*d^7*e^8 - b^5*d^6*e^9)*x^4 + 10*(c^5*d^12*e^3 - 5*b*c^4*d^11*e^4 + 10*b^2*c^3*d^10*e^5 -
10*b^3*c^2*d^9*e^6 + 5*b^4*c*d^8*e^7 - b^5*d^7*e^8)*x^3 + 10*(c^5*d^13*e^2 - 5*b*c^4*d^12*e^3 + 10*b^2*c^3*d^1
1*e^4 - 10*b^3*c^2*d^10*e^5 + 5*b^4*c*d^9*e^6 - b^5*d^8*e^7)*x^2 + 5*(c^5*d^14*e - 5*b*c^4*d^13*e^2 + 10*b^2*c
^3*d^12*e^3 - 10*b^3*c^2*d^11*e^4 + 5*b^4*c*d^10*e^5 - b^5*d^9*e^6)*x), 1/1920*(15*(7*A*b^5*d^5*e^3 + 16*(B*b^
3*c^2 - 2*A*b^2*c^3)*d^8 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^7*e + 3*(B*b^5 - 10*A*b^4*c)*d^6*e^2 + (7*A*b^5*e^8 +
16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^3*e^5 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^2*e^6 + 3*(B*b^5 - 10*A*b^4*c)*d*e^7)*x^5
+ 5*(7*A*b^5*d*e^7 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^4*e^4 - 12*(B*b^4*c - 4*A*b^3*c^2)*d^3*e^5 + 3*(B*b^5 - 10
*A*b^4*c)*d^2*e^6)*x^4 + 10*(7*A*b^5*d^2*e^6 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^5*e^3 - 12*(B*b^4*c - 4*A*b^3*c^
2)*d^4*e^4 + 3*(B*b^5 - 10*A*b^4*c)*d^3*e^5)*x^3 + 10*(7*A*b^5*d^3*e^5 + 16*(B*b^3*c^2 - 2*A*b^2*c^3)*d^6*e^2
- 12*(B*b^4*c - 4*A*b^3*c^2)*d^5*e^3 + 3*(B*b^5 - 10*A*b^4*c)*d^4*e^4)*x^2 + 5*(7*A*b^5*d^4*e^4 + 16*(B*b^3*c^
2 - 2*A*b^2*c^3)*d^7*e - 12*(B*b^4*c - 4*A*b^3*c^2)*d^6*e^2 + 3*(B*b^5 - 10*A*b^4*c)*d^5*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)) + (105*A*b^5*d^5*e^4 - 240*(B*b^2*c^3
- 2*A*b*c^4)*d^9 + 60*(7*B*b^3*c^2 - 20*A*b^2*c^3)*d^8*e - 45*(5*B*b^4*c - 26*A*b^3*c^2)*d^7*e^2 + 15*(3*B*b^5
- 37*A*b^4*c)*d^6*e^3 + (64*B*c^5*d^7*e^2 - 105*A*b^5*d*e^8 - 16*(17*B*b*c^4 - 6*A*c^5)*d^6*e^3 + 32*(11*B*b^
2*c^3 - 9*A*b*c^4)*d^5*e^4 - 2*(147*B*b^3*c^2 - 334*A*b^2*c^3)*d^4*e^5 + (195*B*b^4*c - 856*A*b^3*c^2)*d^3*e^6
- 5*(9*B*b^5 - 97*A*b^4*c)*d^2*e^7)*x^4 + 2*(160*B*c^5*d^8*e - 245*A*b^5*d^2*e^7 - 24*(29*B*b*c^4 - 10*A*c^5)
*d^7*e^2 + 12*(79*B*b^2*c^3 - 62*A*b*c^4)*d^6*e^3 - (763*B*b^3*c^2 - 1622*A*b^2*c^3)*d^5*e^4 + 3*(152*B*b^4*c
- 669*A*b^3*c^2)*d^4*e^5 - 21*(5*B*b^5 - 54*A*b^4*c)*d^3*e^6)*x^3 + 2*(320*B*c^5*d^9 - 448*A*b^5*d^3*e^6 - 480
*(3*B*b*c^4 - A*c^5)*d^8*e + 24*(88*B*b^2*c^3 - 65*A*b*c^4)*d^7*e^2 - (1691*B*b^3*c^2 - 3178*A*b^2*c^3)*d^6*e^
3 + 33*(27*B*b^4*c - 113*A*b^3*c^2)*d^5*e^4 - 3*(64*B*b^5 - 693*A*b^4*c)*d^4*e^5)*x^2 - 10*(79*A*b^5*d^4*e^5 -
16*(B*b*c^4 + 6*A*c^5)*d^9 + 28*(5*B*b^2*c^3 + 12*A*b*c^4)*d^8*e - (211*B*b^3*c^2 + 642*A*b^2*c^3)*d^7*e^2 +
(108*B*b^4*c + 697*A*b^3*c^2)*d^6*e^3 - (21*B*b^5 + 374*A*b^4*c)*d^5*e^4)*x)*sqrt(c*x^2 + b*x))/(c^5*d^15 - 5*
b*c^4*d^14*e + 10*b^2*c^3*d^13*e^2 - 10*b^3*c^2*d^12*e^3 + 5*b^4*c*d^11*e^4 - b^5*d^10*e^5 + (c^5*d^10*e^5 - 5
*b*c^4*d^9*e^6 + 10*b^2*c^3*d^8*e^7 - 10*b^3*c^2*d^7*e^8 + 5*b^4*c*d^6*e^9 - b^5*d^5*e^10)*x^5 + 5*(c^5*d^11*e
^4 - 5*b*c^4*d^10*e^5 + 10*b^2*c^3*d^9*e^6 - 10*b^3*c^2*d^8*e^7 + 5*b^4*c*d^7*e^8 - b^5*d^6*e^9)*x^4 + 10*(c^5
*d^12*e^3 - 5*b*c^4*d^11*e^4 + 10*b^2*c^3*d^10*e^5 - 10*b^3*c^2*d^9*e^6 + 5*b^4*c*d^8*e^7 - b^5*d^7*e^8)*x^3 +
10*(c^5*d^13*e^2 - 5*b*c^4*d^12*e^3 + 10*b^2*c^3*d^11*e^4 - 10*b^3*c^2*d^10*e^5 + 5*b^4*c*d^9*e^6 - b^5*d^8*e
^7)*x^2 + 5*(c^5*d^14*e - 5*b*c^4*d^13*e^2 + 10*b^2*c^3*d^12*e^3 - 10*b^3*c^2*d^11*e^4 + 5*b^4*c*d^10*e^5 - b^
5*d^9*e^6)*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)**(1/2)/(e*x+d)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.59275, size = 5538, normalized size = 12.33 \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)^(1/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

1/128*(16*B*b^3*c^2*d^3 - 32*A*b^2*c^3*d^3 - 12*B*b^4*c*d^2*e + 48*A*b^3*c^2*d^2*e + 3*B*b^5*d*e^2 - 30*A*b^4*
c*d*e^2 + 7*A*b^5*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^4*d^8
- 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 + b^4*d^4*e^4)*sqrt(-c*d^2 + b*d*e)) + 1/1920*(5120*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^6*B*c^(13/2)*d^9*e + 2048*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*c^7*d^10 + 5120*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^7*B*c^6*d^8*e^2 + 3584*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b*c^6*d^9*e + 3072*(sq
rt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^7*d^9*e + 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b*c^(13/2)*d^10 - 8960*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b*c^(11/2)*d^8*e^2 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*c^(13/2)*d^8*
e^2 - 8960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c^(11/2)*d^9*e + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b
*c^(13/2)*d^9*e + 5120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c^6*d^10 - 20480*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^7*B*b*c^5*d^7*e^3 - 24832*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^2*c^5*d^8*e^2 + 9216*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^5*A*b*c^6*d^8*e^2 - 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*c^5*d^9*e + 7680*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^3*A*b^2*c^6*d^9*e + 2560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^3*c^(11/2)*d^10 - 15360*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^6*B*b^2*c^(9/2)*d^7*e^3 - 30720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b*c^(11/2)*d^7*e^3
- 12800*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^3*c^(9/2)*d^8*e^2 - 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^2
*c^(11/2)*d^8*e^2 - 8000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^4*c^(9/2)*d^9*e + 3840*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^2*A*b^3*c^(11/2)*d^9*e + 640*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^4*c^5*d^10 + 30720*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^7*B*b^2*c^4*d^6*e^4 + 13760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^3*c^4*d^7*e^3 - 50048*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^5*A*b^2*c^5*d^7*e^3 + 3200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^4*c^4*d^8*e^2 - 11520
*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*c^5*d^8*e^2 - 2080*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^5*c^4*d^9*e +
960*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*c^5*d^9*e + 64*B*b^5*c^(9/2)*d^10 + 36320*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^6*B*b^3*c^(7/2)*d^6*e^4 + 70720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^(9/2)*d^6*e^4 + 15520*(sqrt(c)
*x - sqrt(c*x^2 + b*x))^4*B*b^4*c^(7/2)*d^7*e^3 - 17600*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^3*c^(9/2)*d^7*e^
3 + 4720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^5*c^(7/2)*d^8*e^2 - 7200*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^
4*c^(9/2)*d^8*e^2 - 208*B*b^6*c^(7/2)*d^9*e + 96*A*b^5*c^(9/2)*d^9*e - 28000*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7
*B*b^3*c^3*d^5*e^5 + 15040*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^2*c^4*d^5*e^5 + 2000*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^5*B*b^4*c^3*d^6*e^4 + 129280*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^3*c^4*d^6*e^4 + 1280*(sqrt(c)*x - s
qrt(c*x^2 + b*x))^3*B*b^5*c^3*d^7*e^3 + 14080*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^4*c^4*d^7*e^3 + 1440*(sqrt
(c)*x - sqrt(c*x^2 + b*x))*B*b^6*c^3*d^8*e^2 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^5*c^4*d^8*e^2 - 2160*(
sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^3*c^(5/2)*d^4*e^6 + 4320*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^2*c^(7/2)*
d^4*e^6 - 39560*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^4*c^(5/2)*d^5*e^5 - 52000*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^6*A*b^3*c^(7/2)*d^5*e^5 - 14360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^5*c^(5/2)*d^6*e^4 + 81920*(sqrt(c)*x -
sqrt(c*x^2 + b*x))^4*A*b^4*c^(7/2)*d^6*e^4 - 3120*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^6*c^(5/2)*d^7*e^3 + 1
3760*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^5*c^(7/2)*d^7*e^3 + 144*B*b^7*c^(5/2)*d^8*e^2 - 192*A*b^6*c^(7/2)*d
^8*e^2 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^3*c^2*d^3*e^7 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^2
*c^3*d^3*e^7 + 9640*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^4*c^2*d^4*e^6 - 20320*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^7*A*b^3*c^3*d^4*e^6 - 17284*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^5*c^2*d^5*e^5 - 120680*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^5*A*b^4*c^3*d^5*e^5 - 6920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^6*c^2*d^6*e^4 + 14080*(sqrt(c)*x
- sqrt(c*x^2 + b*x))^3*A*b^5*c^3*d^6*e^4 - 1260*(sqrt(c)*x - sqrt(c*x^2 + b*x))*B*b^7*c^2*d^7*e^3 + 4280*(sqr
t(c)*x - sqrt(c*x^2 + b*x))*A*b^6*c^3*d^7*e^3 + 1620*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b^4*c^(3/2)*d^3*e^7 -
6480*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^3*c^(5/2)*d^3*e^7 + 15090*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*b^5*
c^(3/2)*d^4*e^6 + 7260*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^4*c^(5/2)*d^4*e^6 + 330*(sqrt(c)*x - sqrt(c*x^2 +
b*x))^4*B*b^6*c^(3/2)*d^5*e^5 - 85780*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^5*c^(5/2)*d^5*e^5 - 570*(sqrt(c)*
x - sqrt(c*x^2 + b*x))^2*B*b^7*c^(3/2)*d^6*e^4 - 6340*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^6*c^(5/2)*d^6*e^4
- 150*B*b^8*c^(3/2)*d^7*e^3 + 476*A*b^7*c^(5/2)*d^7*e^3 + 180*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b^4*c*d^2*e^
8 - 720*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^3*c^2*d^2*e^8 - 570*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^5*c*d^
3*e^7 + 10740*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^4*c^2*d^3*e^7 + 7878*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b
^6*c*d^4*e^6 + 47944*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^5*c^2*d^4*e^6 + 2370*(sqrt(c)*x - sqrt(c*x^2 + b*x)
)^3*B*b^7*c*d^5*e^5 - 25220*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^6*c^2*d^5*e^5 + 270*(sqrt(c)*x - sqrt(c*x^2
+ b*x))*B*b^8*c*d^6*e^4 - 3080*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^7*c^2*d^6*e^4 - 405*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^8*B*b^5*sqrt(c)*d^2*e^8 + 4050*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*b^4*c^(3/2)*d^2*e^8 - 1470*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^6*B*b^6*sqrt(c)*d^3*e^7 + 9310*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^5*c^(3/2)*d^3*e^
7 + 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^7*sqrt(c)*d^4*e^6 + 35330*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b
^6*c^(3/2)*d^4*e^6 + 630*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^8*sqrt(c)*d^5*e^5 - 1750*(sqrt(c)*x - sqrt(c*x^
2 + b*x))^2*A*b^7*c^(3/2)*d^5*e^5 + 45*B*b^9*sqrt(c)*d^6*e^4 - 380*A*b^8*c^(3/2)*d^6*e^4 - 45*(sqrt(c)*x - sqr
t(c*x^2 + b*x))^9*B*b^5*d*e^9 + 450*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^4*c*d*e^9 - 210*(sqrt(c)*x - sqrt(c*
x^2 + b*x))^7*B*b^6*d^2*e^8 - 1190*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b^5*c*d^2*e^8 - 384*(sqrt(c)*x - sqrt(c
*x^2 + b*x))^5*B*b^7*d^3*e^7 - 4658*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^6*c*d^3*e^7 + 210*(sqrt(c)*x - sqrt(
c*x^2 + b*x))^3*B*b^8*d^4*e^6 + 10510*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^7*c*d^4*e^6 + 45*(sqrt(c)*x - sqrt
(c*x^2 + b*x))*B*b^9*d^5*e^5 + 600*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^8*c*d^5*e^5 - 945*(sqrt(c)*x - sqrt(c*x
^2 + b*x))^8*A*b^5*sqrt(c)*d*e^9 - 3430*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^6*sqrt(c)*d^2*e^8 - 4480*(sqrt(c
)*x - sqrt(c*x^2 + b*x))^4*A*b^7*sqrt(c)*d^3*e^7 + 1470*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*A*b^8*sqrt(c)*d^4*e^
6 + 105*A*b^9*sqrt(c)*d^5*e^5 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b^5*e^10 - 490*(sqrt(c)*x - sqrt(c*x^2
+ b*x))^7*A*b^6*d*e^9 - 896*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*b^7*d^2*e^8 - 790*(sqrt(c)*x - sqrt(c*x^2 + b
*x))^3*A*b^8*d^3*e^7 + 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^9*d^4*e^6)/((c^4*d^8*e^3 - 4*b*c^3*d^7*e^4 + 6*
b^2*c^2*d^6*e^5 - 4*b^3*c*d^5*e^6 + b^4*d^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)^5)