3.1248 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=363 \[ \frac{\sqrt{d+e x} \left (x \left (b^2 c^2 d e (14 A e+11 B d)-A b^3 c e^3-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-3 b^4 B e^3\right )+b c d^2 \left (-b c (11 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac{(c d-b e)^{3/2} \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}-\frac{(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \]

[Out]

-((d + e*x)^(5/2)*(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*c*d^2*(12*A*c^2*d + 2*b^2*B*e - b*c*(6*B*d + 11*A*e)) + (24*A*c^4*d^3 - 3*b^4*B*e^3 - A*b^3*c*e^3 - 1
2*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(11*B*d + 14*A*e))*x))/(4*b^4*c^2*(b*x + c*x^2)) - (d^(3/2)*(48*A*c^2*
d^2 + 7*b^2*e*(4*B*d + 5*A*e) - 12*b*c*d*(2*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b*
e)^(3/2)*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*(8*B*d + A*e))*ArcTanh[(Sqrt[c]*Sqrt
[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.826888, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {818, 826, 1166, 208} \[ \frac{\sqrt{d+e x} \left (x \left (b^2 c^2 d e (14 A e+11 B d)-A b^3 c e^3-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-3 b^4 B e^3\right )+b c d^2 \left (-b c (11 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac{(c d-b e)^{3/2} \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}-\frac{(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x]

[Out]

-((d + e*x)^(5/2)*(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*c*d^2*(12*A*c^2*d + 2*b^2*B*e - b*c*(6*B*d + 11*A*e)) + (24*A*c^4*d^3 - 3*b^4*B*e^3 - A*b^3*c*e^3 - 1
2*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(11*B*d + 14*A*e))*x))/(4*b^4*c^2*(b*x + c*x^2)) - (d^(3/2)*(48*A*c^2*
d^2 + 7*b^2*e*(4*B*d + 5*A*e) - 12*b*c*d*(2*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b*
e)^(3/2)*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*(8*B*d + A*e))*ArcTanh[(Sqrt[c]*Sqrt
[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))

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 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)^{7/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{5/2} \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{(d+e x)^{3/2} \left (-\frac{1}{2} d \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )-\frac{1}{2} e \left (2 A c^2 d-3 b^2 B e-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)^{5/2} \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 c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac{\int \frac{\frac{1}{4} c^2 d^2 \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right )+\frac{1}{4} e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c^2}\\ &=-\frac{(d+e x)^{5/2} \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 c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{4} c^2 d^2 e \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right )-\frac{1}{4} d e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 A e)\right )+\frac{1}{4} e \left (24 A c^4 d^3+3 b^4 B e^3+b^3 c e^2 (2 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+10 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^2}\\ &=-\frac{(d+e x)^{5/2} \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 c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac{\left ((c d-b e)^2 \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+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^2}+\frac{\left (c d^2 \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 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)^{5/2} \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 c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+11 A e)\right )+\left (24 A c^4 d^3-3 b^4 B e^3-A b^3 c e^3-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (11 B d+14 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac{d^{3/2} \left (48 A c^2 d^2+7 b^2 e (4 B d+5 A e)-12 b c d (2 B d+7 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{(c d-b e)^{3/2} \left (48 A c^3 d^2-3 b^3 B e^2-12 b c^2 d (2 B d+A e)-b^2 c e (8 B d+A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 4.54039, size = 554, normalized size = 1.53 \[ \frac{\frac{(b+c x) \left ((b+c x) \left (105 c^{9/2} (c d-b e)^2 \left (\frac{2}{15} d \sqrt{d+e x} \left (23 d^2+11 d e x+3 e^2 x^2\right )-2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+\frac{2}{7} (d+e x)^{7/2}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )-2 c^2 d^2 \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt{c} \sqrt{d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )\right )-210 b c^{11/2} (d+e x)^{9/2} \left (-11 b^2 c d e (2 A e+B d)+b^3 e^2 (5 A e+4 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )\right )}{b^4 c^{9/2} d (c d-b e)^2}-\frac{210 c (d+e x)^{9/2} \left (b^2 e (5 A e+4 B d)-3 b c d (5 A e+2 B d)+12 A c^2 d^2\right )}{b^2 d (b e-c d)}-\frac{210 (d+e x)^{9/2} (5 A b e-8 A c d+4 b B d)}{b d x}-\frac{420 A (d+e x)^{9/2}}{x^2}}{840 b d (b+c x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^3,x]

[Out]

((-210*c*(12*A*c^2*d^2 - 3*b*c*d*(2*B*d + 5*A*e) + b^2*e*(4*B*d + 5*A*e))*(d + e*x)^(9/2))/(b^2*d*(-(c*d) + b*
e)) - (420*A*(d + e*x)^(9/2))/x^2 - (210*(4*b*B*d - 8*A*c*d + 5*A*b*e)*(d + e*x)^(9/2))/(b*d*x) + ((b + c*x)*(
-210*b*c^(11/2)*(-24*A*c^3*d^3 - 11*b^2*c*d*e*(B*d + 2*A*e) + 12*b*c^2*d^2*(B*d + 3*A*e) + b^3*e^2*(4*B*d + 5*
A*e))*(d + e*x)^(9/2) + (b + c*x)*(105*c^(9/2)*(c*d - b*e)^2*(48*A*c^2*d^2 + 7*b^2*e*(4*B*d + 5*A*e) - 12*b*c*
d*(2*B*d + 7*A*e))*((2*(d + e*x)^(7/2))/7 + (2*d*Sqrt[d + e*x]*(23*d^2 + 11*d*e*x + 3*e^2*x^2))/15 - 2*d^(7/2)
*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - 2*c^2*d^2*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*
(8*B*d + A*e))*(15*c^(7/2)*(d + e*x)^(7/2) + 7*(c*d - b*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^(9/2)*d*(c*d - b*e)^2))/(840*b*d*(b + c*x)^2)

________________________________________________________________________________________

Maple [B]  time = 0.024, size = 1218, normalized size = 3.4 \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)^(7/2)/(c*x^2+b*x)^3,x)

[Out]

21*e*d^(5/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c+6/b^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d
)*c)^(1/2))*B*d^4*c^2-12/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^4+15/4*e^
3/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*B*d^2+5/2*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*d-3/4*e^5*b/(c*e*x+b*e)^2/c^2*(e*x+d)^(1/2)*B+1/4*e^4/b/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b
*e-c*d)*c)^(1/2))*A+1/2*e^3/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d+11/4*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^2+15/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A*d-1/e*d^3/b^3/x^2*(e*x+d)^(3/2)*B+
1/e*d^4/b^3/x^2*(e*x+d)^(1/2)*B+1/4*e^4/(c*e*x+b*e)^2/c*(e*x+d)^(1/2)*B*d+2*e/b^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*
B*d^4*c^2+5/2*e^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c*d-23/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^2*d^2+3*e
/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^3*d^3+11/4*e^2/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(3/2)*B*d^2+3/e*d^3/b^4/x^2*(e
*x+d)^(3/2)*A*c-3/e*d^4/b^4/x^2*(e*x+d)^(1/2)*A*c+6*d^(7/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c-13/4*d^2/b^
3/x^2*(e*x+d)^(3/2)*A+11/4*d^3/b^3/x^2*(e*x+d)^(1/2)*A-12*d^(7/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2-5/4
*e^4/(c*e*x+b*e)^2/c*(e*x+d)^(3/2)*B-1/4*e^5/(c*e*x+b*e)^2/c*(e*x+d)^(1/2)*A+3/4*e^4/c^2/((b*e-c*d)*c)^(1/2)*a
rctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B+1/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A-35/4*e^2*d^(3/2)/b^3*arct
anh((e*x+d)^(1/2)/d^(1/2))*A-7*e*d^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*B-2*e/b^3/(c*e*x+b*e)^2*(e*x+d)^(3
/2)*B*d^3*c^2-71/4*e^2/b^3*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^2+27*e/b^4/((
b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^3*c^2+1/2*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*d-10*e/b^3*c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^
(1/2))*B*d^3-39/4*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*A*d^2+37/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A*d^3*c
^2-3*e/b^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A*c^3*d^4-21/4*e^2/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*B*d^3

________________________________________________________________________________________

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)^(7/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [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)^(7/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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)**(7/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.49238, size = 1413, normalized size = 3.89 \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)^(7/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-1/4*(24*B*b*c*d^4 - 48*A*c^2*d^4 - 28*B*b^2*d^3*e + 84*A*b*c*d^3*e - 35*A*b^2*d^2*e^2)*arctan(sqrt(x*e + d)/s
qrt(-d))/(b^5*sqrt(-d)) + 1/4*(24*B*b*c^4*d^4 - 48*A*c^5*d^4 - 40*B*b^2*c^3*d^3*e + 108*A*b*c^4*d^3*e + 11*B*b
^3*c^2*d^2*e^2 - 71*A*b^2*c^3*d^2*e^2 + 2*B*b^4*c*d*e^3 + 10*A*b^3*c^2*d*e^3 + 3*B*b^5*e^4 + A*b^4*c*e^4)*arct
an(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e)*b^5*c^2) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^4*d^3*
e - 24*(x*e + d)^(7/2)*A*c^5*d^3*e - 36*(x*e + d)^(5/2)*B*b*c^4*d^4*e + 72*(x*e + d)^(5/2)*A*c^5*d^4*e + 36*(x
*e + d)^(3/2)*B*b*c^4*d^5*e - 72*(x*e + d)^(3/2)*A*c^5*d^5*e - 12*sqrt(x*e + d)*B*b*c^4*d^6*e + 24*sqrt(x*e +
d)*A*c^5*d^6*e - 11*(x*e + d)^(7/2)*B*b^2*c^3*d^2*e^2 + 36*(x*e + d)^(7/2)*A*b*c^4*d^2*e^2 + 51*(x*e + d)^(5/2
)*B*b^2*c^3*d^3*e^2 - 144*(x*e + d)^(5/2)*A*b*c^4*d^3*e^2 - 69*(x*e + d)^(3/2)*B*b^2*c^3*d^4*e^2 + 180*(x*e +
d)^(3/2)*A*b*c^4*d^4*e^2 + 29*sqrt(x*e + d)*B*b^2*c^3*d^5*e^2 - 72*sqrt(x*e + d)*A*b*c^4*d^5*e^2 - 2*(x*e + d)
^(7/2)*B*b^3*c^2*d*e^3 - 10*(x*e + d)^(7/2)*A*b^2*c^3*d*e^3 - 11*(x*e + d)^(5/2)*B*b^3*c^2*d^2*e^3 + 85*(x*e +
d)^(5/2)*A*b^2*c^3*d^2*e^3 + 32*(x*e + d)^(3/2)*B*b^3*c^2*d^3*e^3 - 148*(x*e + d)^(3/2)*A*b^2*c^3*d^3*e^3 - 1
9*sqrt(x*e + d)*B*b^3*c^2*d^4*e^3 + 73*sqrt(x*e + d)*A*b^2*c^3*d^4*e^3 + 5*(x*e + d)^(7/2)*B*b^4*c*e^4 - (x*e
+ d)^(7/2)*A*b^3*c^2*e^4 - 11*(x*e + d)^(5/2)*B*b^4*c*d*e^4 - 13*(x*e + d)^(5/2)*A*b^3*c^2*d*e^4 + 7*(x*e + d)
^(3/2)*B*b^4*c*d^2*e^4 + 42*(x*e + d)^(3/2)*A*b^3*c^2*d^2*e^4 - sqrt(x*e + d)*B*b^4*c*d^3*e^4 - 26*sqrt(x*e +
d)*A*b^3*c^2*d^3*e^4 + 3*(x*e + d)^(5/2)*B*b^5*e^5 + (x*e + d)^(5/2)*A*b^4*c*e^5 - 6*(x*e + d)^(3/2)*B*b^5*d*e
^5 - 2*(x*e + d)^(3/2)*A*b^4*c*d*e^5 + 3*sqrt(x*e + d)*B*b^5*d^2*e^5 + sqrt(x*e + d)*A*b^4*c*d^2*e^5)/(((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^2)