3.1249 \(\int \frac{(A+B x) (d+e x)^{5/2}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=344 \[ -\frac{\sqrt{d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{4 b^4 c \left (b x+c x^2\right )}+\frac{\sqrt{c d-b e} \left (b^2 c e (3 A e+8 B d)-12 b c^2 d (3 A e+2 B d)+48 A c^3 d^2+b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{3/2}}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (5 b^2 e (3 A e+4 B d)-12 b c d (5 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}-\frac{(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \]

[Out]

-((d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(2*b^2*c*(b*x + c*x^2)^2) - (Sqrt[d +
e*x]*(b*d*(6*b*B*c*d - 12*A*c^2*d - 2*b^2*B*e + 9*A*b*c*e) - (24*A*c^3*d^2 + b^3*B*e^2 - 12*b*c^2*d*(B*d + 2*
A*e) + b^2*c*e*(7*B*d + 3*A*e))*x))/(4*b^4*c*(b*x + c*x^2)) - (Sqrt[d]*(48*A*c^2*d^2 + 5*b^2*e*(4*B*d + 3*A*e)
- 12*b*c*d*(2*B*d + 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (Sqrt[c*d - b*e]*(48*A*c^3*d^2 + b^3*B*
e^2 - 12*b*c^2*d*(2*B*d + 3*A*e) + b^2*c*e*(8*B*d + 3*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/
(4*b^5*c^(3/2))

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Rubi [A]  time = 0.697501, antiderivative size = 344, 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 = {818, 820, 826, 1166, 208} \[ -\frac{\sqrt{d+e x} \left (b d \left (9 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )-x \left (b^2 c e (3 A e+7 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+b^3 B e^2\right )\right )}{4 b^4 c \left (b x+c x^2\right )}+\frac{\sqrt{c d-b e} \left (b^2 c e (3 A e+8 B d)-12 b c^2 d (3 A e+2 B d)+48 A c^3 d^2+b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{3/2}}-\frac{\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (5 b^2 e (3 A e+4 B d)-12 b c d (5 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}-\frac{(d+e x)^{3/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d + e*x)^(3/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(2*b^2*c*(b*x + c*x^2)^2) - (Sqrt[d +
e*x]*(b*d*(6*b*B*c*d - 12*A*c^2*d - 2*b^2*B*e + 9*A*b*c*e) - (24*A*c^3*d^2 + b^3*B*e^2 - 12*b*c^2*d*(B*d + 2*
A*e) + b^2*c*e*(7*B*d + 3*A*e))*x))/(4*b^4*c*(b*x + c*x^2)) - (Sqrt[d]*(48*A*c^2*d^2 + 5*b^2*e*(4*B*d + 3*A*e)
- 12*b*c*d*(2*B*d + 5*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (Sqrt[c*d - b*e]*(48*A*c^3*d^2 + b^3*B*
e^2 - 12*b*c^2*d*(2*B*d + 3*A*e) + b^2*c*e*(8*B*d + 3*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/
(4*b^5*c^(3/2))

Rule 818

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 + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*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] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p]
|| IntegersQ[2*m, 2*p])

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*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]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^{5/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac{\int \frac{\sqrt{d+e x} \left (\frac{1}{2} d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\frac{1}{2} e \left (6 A c^2 d-b^2 B e-3 b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2 c}\\ &=-\frac{(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (b d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{\int \frac{-\frac{1}{4} c d \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 A e)\right )-\frac{1}{4} e \left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c}\\ &=-\frac{(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (b d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} d e \left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right )-\frac{1}{4} c d e \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 A e)\right )-\frac{1}{4} e \left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{b^4 c}\\ &=-\frac{(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (b d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{\left ((c d-b e) \left (48 A c^3 d^2+b^3 B e^2-12 b c^2 d (2 B d+3 A e)+b^2 c e (8 B d+3 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5 c}+\frac{\left (c d \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5}\\ &=-\frac{(d+e x)^{3/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}-\frac{\sqrt{d+e x} \left (b d \left (6 b B c d-12 A c^2 d-2 b^2 B e+9 A b c e\right )-\left (24 A c^3 d^2+b^3 B e^2-12 b c^2 d (B d+2 A e)+b^2 c e (7 B d+3 A e)\right ) x\right )}{4 b^4 c \left (b x+c x^2\right )}-\frac{\sqrt{d} \left (48 A c^2 d^2+5 b^2 e (4 B d+3 A e)-12 b c d (2 B d+5 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{\sqrt{c d-b e} \left (48 A c^3 d^2+b^3 B e^2-12 b c^2 d (2 B d+3 A e)+b^2 c e (8 B d+3 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 3.38293, size = 516, normalized size = 1.5 \[ \frac{\frac{(b+c x) \left ((b+c x) \left (15 c^{7/2} (c d-b e)^2 \left (-2 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+\frac{2}{3} d \sqrt{d+e x} (4 d+e x)+\frac{2}{5} (d+e x)^{5/2}\right ) \left (5 b^2 e (3 A e+4 B d)-12 b c d (5 A e+2 B d)+48 A c^2 d^2\right )-2 c^2 d^2 \left (b^2 c e (3 A e+8 B d)-12 b c^2 d (3 A e+2 B d)+48 A c^3 d^2+b^3 B e^2\right ) \left (5 (c d-b e) \left (\sqrt{c} \sqrt{d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )\right )-30 b c^{9/2} (d+e x)^{7/2} \left (-b^2 c d e (18 A e+13 B d)+b^3 e^2 (3 A e+4 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )\right )}{b^4 c^{7/2} d (c d-b e)^2}+\frac{30 c (d+e x)^{7/2} \left (b^2 (-e) (3 A e+4 B d)+b c d (13 A e+6 B d)-12 A c^2 d^2\right )}{b^2 d (b e-c d)}-\frac{30 (d+e x)^{7/2} (3 A b e-8 A c d+4 b B d)}{b d x}-\frac{60 A (d+e x)^{7/2}}{x^2}}{120 b d (b+c x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((30*c*(-12*A*c^2*d^2 - b^2*e*(4*B*d + 3*A*e) + b*c*d*(6*B*d + 13*A*e))*(d + e*x)^(7/2))/(b^2*d*(-(c*d) + b*e)
) - (60*A*(d + e*x)^(7/2))/x^2 - (30*(4*b*B*d - 8*A*c*d + 3*A*b*e)*(d + e*x)^(7/2))/(b*d*x) + ((b + c*x)*(-30*
b*c^(9/2)*(-24*A*c^3*d^3 + 12*b*c^2*d^2*(B*d + 3*A*e) + b^3*e^2*(4*B*d + 3*A*e) - b^2*c*d*e*(13*B*d + 18*A*e))
*(d + e*x)^(7/2) + (b + c*x)*(15*c^(7/2)*(c*d - b*e)^2*(48*A*c^2*d^2 + 5*b^2*e*(4*B*d + 3*A*e) - 12*b*c*d*(2*B
*d + 5*A*e))*((2*(d + e*x)^(5/2))/5 + (2*d*Sqrt[d + e*x]*(4*d + e*x))/3 - 2*d^(5/2)*ArcTanh[Sqrt[d + e*x]/Sqrt
[d]]) - 2*c^2*d^2*(48*A*c^3*d^2 + b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 3*A*e) + b^2*c*e*(8*B*d + 3*A*e))*(3*c^(5/2)
*(d + e*x)^(5/2) + 5*(c*d - b*e)*(Sqrt[c]*Sqrt[d + e*x]*(4*c*d - 3*b*e + c*e*x) - 3*(c*d - b*e)^(3/2)*ArcTanh[
(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])))))/(b^4*c^(7/2)*d*(c*d - b*e)^2))/(120*b*d*(b + c*x)^2)

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Maple [B]  time = 0.023, size = 1004, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/b^4*c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^3-12/b^5*c^3/((b*e-c*d)*c)^(1/2)
*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^3+15*e*d^(3/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c+5/2*e^3
/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*B*d+7/4*e^2/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))
*B*d+3/4*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c+1/4*e^3/b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-
c*d)*c)^(1/2))*B+1/e*d^3/b^3/x^2*(e*x+d)^(1/2)*B-1/e*d^2/b^3/x^2*(e*x+d)^(3/2)*B+6*d^(5/2)/b^4*arctanh((e*x+d)
^(1/2)/d^(1/2))*B*c+7/4*d^2/b^3/x^2*(e*x+d)^(1/2)*A-9/4*d/b^3/x^2*(e*x+d)^(3/2)*A-1/4*e^4/(c*e*x+b*e)^2/c*(e*x
+d)^(1/2)*B+3/4*e^3/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A+1/4*e^3/b/(c*e*x+b*e
)^2*(e*x+d)^(3/2)*B+5/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A-15/4*e^2*d^(1/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2)
)*A-5*e*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*B-12*d^(5/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2-39/4*
e^2/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*c*d+21*e/b^4*c^2/((b*e-c*d)*c)^(1/2)
*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^2+3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^3*d^2+3/e*d^2/b^4/x
^2*(e*x+d)^(3/2)*A*c-3/e*d^3/b^4/x^2*(e*x+d)^(1/2)*A*c-8*e/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*
e-c*d)*c)^(1/2))*B*d^2*c-15/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^2*d+7/4*e^2/b^2/(c*e*x+b*e)^2*(e*x+d)^(3
/2)*B*c*d-2*e/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*c^2*d^2-11/2*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A*c*d+29/4*e^
2/b^3/(c*e*x+b*e)^2*c^2*(e*x+d)^(1/2)*A*d^2-3*e/b^4/(c*e*x+b*e)^2*c^3*(e*x+d)^(1/2)*A*d^3-17/4*e^2/b^2/(c*e*x+
b*e)^2*(e*x+d)^(1/2)*B*d^2*c+2*e/b^3/(c*e*x+b*e)^2*c^2*(e*x+d)^(1/2)*B*d^3

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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)*(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 31.2012, size = 5739, normalized size = 16.68 \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)*(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(((24*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3*c^2 + 3*A*b^2*c^3)*e^2)*x^4 + 2
*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (24*
(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A*b^4*c)*e^2)*x^2)*sqrt((c*d - b*
e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - ((15*A*b^2*c^3*e^2 - 24*(
B*b*c^4 - 2*A*c^5)*d^2 + 20*(B*b^2*c^3 - 3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4
)*d^2 + 20*(B*b^3*c^2 - 3*A*b^2*c^3)*d*e)*x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4
*c - 3*A*b^3*c^2)*d*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c*d^2 + (12*(B*b
^2*c^3 - 2*A*b*c^4)*d^2 - (7*B*b^3*c^2 - 24*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2
- 2*A*b^2*c^3)*d^2 - (11*B*b^4*c - 37*A*b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*e^2)*x^2 + (9*A*b^4*c*d*e + 4*(B*b
^4*c - 2*A*b^3*c^2)*d^2)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), -1/8*(2*((24*(B*b*c^4 -
2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3*c^2 + 3*A*b^2*c^3)*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b
*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^
3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A*b^4*c)*e^2)*x^2)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x
+ d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((15*A*b^2*c^3*e^2 - 24*(B*b*c^4 - 2*A*c^5)*d^2 + 20*(B*b^2*c^3 -
3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 + 20*(B*b^3*c^2 - 3*A*b^2*c^3)*d*e)
*x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4*c - 3*A*b^3*c^2)*d*e)*x^2)*sqrt(d)*log((
e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c*d^2 + (12*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - (7*B*b^3*c^2 -
24*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - (11*B*b^4*c - 37*A*
b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*e^2)*x^2 + (9*A*b^4*c*d*e + 4*(B*b^4*c - 2*A*b^3*c^2)*d^2)*x)*sqrt(e*x + d)
)/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2), 1/8*(2*((15*A*b^2*c^3*e^2 - 24*(B*b*c^4 - 2*A*c^5)*d^2 + 20*(B*b^
2*c^3 - 3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 + 20*(B*b^3*c^2 - 3*A*b^2*c
^3)*d*e)*x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4*c - 3*A*b^3*c^2)*d*e)*x^2)*sqrt(
-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - ((24*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3
*c^2 + 3*A*b^2*c^3)*e^2)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*
c + 3*A*b^3*c^2)*e^2)*x^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A
*b^4*c)*e^2)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x +
b)) - 2*(2*A*b^4*c*d^2 + (12*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - (7*B*b^3*c^2 - 24*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*
b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - (11*B*b^4*c - 37*A*b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*
e^2)*x^2 + (9*A*b^4*c*d*e + 4*(B*b^4*c - 2*A*b^3*c^2)*d^2)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^
7*c*x^2), -1/4*(((24*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(2*B*b^2*c^3 - 9*A*b*c^4)*d*e - (B*b^3*c^2 + 3*A*b^2*c^3)*e^2
)*x^4 + 2*(24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(2*B*b^3*c^2 - 9*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x
^3 + (24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 - 4*(2*B*b^4*c - 9*A*b^3*c^2)*d*e - (B*b^5 + 3*A*b^4*c)*e^2)*x^2)*sqrt(
-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) - ((15*A*b^2*c^3*e^2 - 24*(B*b*c^4 -
2*A*c^5)*d^2 + 20*(B*b^2*c^3 - 3*A*b*c^4)*d*e)*x^4 + 2*(15*A*b^3*c^2*e^2 - 24*(B*b^2*c^3 - 2*A*b*c^4)*d^2 + 2
0*(B*b^3*c^2 - 3*A*b^2*c^3)*d*e)*x^3 + (15*A*b^4*c*e^2 - 24*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 + 20*(B*b^4*c - 3*A*
b^3*c^2)*d*e)*x^2)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (2*A*b^4*c*d^2 + (12*(B*b^2*c^3 - 2*A*b*c^4)*d^
2 - (7*B*b^3*c^2 - 24*A*b^2*c^3)*d*e - (B*b^4*c + 3*A*b^3*c^2)*e^2)*x^3 + (18*(B*b^3*c^2 - 2*A*b^2*c^3)*d^2 -
(11*B*b^4*c - 37*A*b^3*c^2)*d*e + (B*b^5 - 5*A*b^4*c)*e^2)*x^2 + (9*A*b^4*c*d*e + 4*(B*b^4*c - 2*A*b^3*c^2)*d^
2)*x)*sqrt(e*x + d))/(b^5*c^3*x^4 + 2*b^6*c^2*x^3 + b^7*c*x^2)]

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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)*(e*x+d)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.57058, size = 1135, normalized size = 3.3 \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)*(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-1/4*(24*B*b*c*d^3 - 48*A*c^2*d^3 - 20*B*b^2*d^2*e + 60*A*b*c*d^2*e - 15*A*b^2*d*e^2)*arctan(sqrt(x*e + d)/sqr
t(-d))/(b^5*sqrt(-d)) + 1/4*(24*B*b*c^3*d^3 - 48*A*c^4*d^3 - 32*B*b^2*c^2*d^2*e + 84*A*b*c^3*d^2*e + 7*B*b^3*c
*d*e^2 - 39*A*b^2*c^2*d*e^2 + B*b^4*e^3 + 3*A*b^3*c*e^3)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c
^2*d + b*c*e)*b^5*c) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^3*d^2*e - 24*(x*e + d)^(7/2)*A*c^4*d^2*e - 36*(x*e + d)^(
5/2)*B*b*c^3*d^3*e + 72*(x*e + d)^(5/2)*A*c^4*d^3*e + 36*(x*e + d)^(3/2)*B*b*c^3*d^4*e - 72*(x*e + d)^(3/2)*A*
c^4*d^4*e - 12*sqrt(x*e + d)*B*b*c^3*d^5*e + 24*sqrt(x*e + d)*A*c^4*d^5*e - 7*(x*e + d)^(7/2)*B*b^2*c^2*d*e^2
+ 24*(x*e + d)^(7/2)*A*b*c^3*d*e^2 + 39*(x*e + d)^(5/2)*B*b^2*c^2*d^2*e^2 - 108*(x*e + d)^(5/2)*A*b*c^3*d^2*e^
2 - 57*(x*e + d)^(3/2)*B*b^2*c^2*d^3*e^2 + 144*(x*e + d)^(3/2)*A*b*c^3*d^3*e^2 + 25*sqrt(x*e + d)*B*b^2*c^2*d^
4*e^2 - 60*sqrt(x*e + d)*A*b*c^3*d^4*e^2 - (x*e + d)^(7/2)*B*b^3*c*e^3 - 3*(x*e + d)^(7/2)*A*b^2*c^2*e^3 - 8*(
x*e + d)^(5/2)*B*b^3*c*d*e^3 + 46*(x*e + d)^(5/2)*A*b^2*c^2*d*e^3 + 23*(x*e + d)^(3/2)*B*b^3*c*d^2*e^3 - 91*(x
*e + d)^(3/2)*A*b^2*c^2*d^2*e^3 - 14*sqrt(x*e + d)*B*b^3*c*d^3*e^3 + 48*sqrt(x*e + d)*A*b^2*c^2*d^3*e^3 + (x*e
+ d)^(5/2)*B*b^4*e^4 - 5*(x*e + d)^(5/2)*A*b^3*c*e^4 - 2*(x*e + d)^(3/2)*B*b^4*d*e^4 + 19*(x*e + d)^(3/2)*A*b
^3*c*d*e^4 + sqrt(x*e + d)*B*b^4*d^2*e^4 - 12*sqrt(x*e + d)*A*b^3*c*d^2*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d
+ c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c)