3.1239 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=292 \[ \frac{e \sqrt{d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{b^2 c^3}-\frac{(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac{e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 b^2 c^2}+\frac{(c d-b e)^{5/2} \left (-3 A b c e-4 A c^2 d+5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}-\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (7 A b e-4 A c d+2 b B d)}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.882068, antiderivative size = 292, 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 = {818, 824, 826, 1166, 208} \[ \frac{e \sqrt{d+e x} \left (b^2 c e (3 A e+8 B d)-b c^2 d (2 A e+B d)+2 A c^3 d^2-5 b^3 B e^2\right )}{b^2 c^3}-\frac{(d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac{e (d+e x)^{3/2} \left (-3 b c (A e+B d)+6 A c^2 d+5 b^2 B e\right )}{3 b^2 c^2}+\frac{(c d-b e)^{5/2} \left (-3 A b c e-4 A c^2 d+5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}-\frac{d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (7 A b e-4 A c d+2 b B d)}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), 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] && FractionQ[m] && GtQ[m, 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 )^2} \, 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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\int \frac{(d+e x)^{3/2} \left (\frac{1}{2} c d (2 b B d-4 A c d+7 A b e)+\frac{1}{2} e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac{e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\int \frac{\sqrt{d+e x} \left (\frac{1}{2} c^2 d^2 (2 b B d-4 A c d+7 A b e)+\frac{1}{2} e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac{e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt{d+e x}}{b^2 c^3}+\frac{e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\int \frac{\frac{1}{2} c^3 d^3 (2 b B d-4 A c d+7 A b e)-\frac{1}{2} e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^3}\\ &=\frac{e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt{d+e x}}{b^2 c^3}+\frac{e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 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 )}{b^2 c \left (b x+c x^2\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} c^3 d^3 e (2 b B d-4 A c d+7 A b e)+\frac{1}{2} d e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 A e)\right )-\frac{1}{2} e \left (2 A c^4 d^3-5 b^4 B e^3-b c^3 d^2 (B d+3 A e)+b^3 c e^2 (13 B d+3 A e)-b^2 c^2 d e (9 B d+5 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^2 c^3}\\ &=\frac{e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt{d+e x}}{b^2 c^3}+\frac{e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\left (c d^3 (2 b B d-4 A c d+7 A b e)\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 )}{b^3}+\frac{\left ((c d-b e)^3 \left (4 A c^2 d-5 b^2 B e-b c (2 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 )}{b^3 c^3}\\ &=\frac{e \left (2 A c^3 d^2-5 b^3 B e^2-b c^2 d (B d+2 A e)+b^2 c e (8 B d+3 A e)\right ) \sqrt{d+e x}}{b^2 c^3}+\frac{e \left (6 A c^2 d+5 b^2 B e-3 b c (B d+A e)\right ) (d+e x)^{3/2}}{3 b^2 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 )}{b^2 c \left (b x+c x^2\right )}-\frac{d^{5/2} (2 b B d-4 A c d+7 A b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}+\frac{(c d-b e)^{5/2} \left (2 b B c d-4 A c^2 d+5 b^2 B e-3 A b c e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 1.7242, size = 341, normalized size = 1.17 \[ -\frac{-\frac{105 \left (\frac{2}{15} d \sqrt{d+e x} \left (23 d^2+11 d e x+3 e^2 x^2\right )-2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )+\frac{2}{7} (d+e x)^{7/2}\right ) (7 A b e-4 A c d+2 b B d)-\frac{2 d \left (b c (2 B d-3 A e)-4 A c^2 d+5 b^2 B e\right ) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt{c} \sqrt{d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )}{c^{7/2} (c d-b e)}}{210 b^2}+\frac{c (d+e x)^{9/2} (A b e-2 A c d+b B d)}{b (b+c x) (b e-c d)}+\frac{A (d+e x)^{9/2}}{x (b+c x)}}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c*(b*B*d - 2*A*c*d + A*b*e)*(d + e*x)^(9/2))/(b*(-(c*d) + b*e)*(b + c*x)) + (A*(d + e*x)^(9/2))/(x*(b + c*
x)) - (105*(2*b*B*d - 4*A*c*d + 7*A*b*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*d*(-4*A*c^2*d + 5*b^2*B*e + b*c*(2*B*d - 3*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]]))))/(c^(7/2)*(c*
d - b*e)))/(210*b^2))/(b*d))

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Maple [B]  time = 0.028, size = 823, normalized size = 2.8 \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)^2,x)

[Out]

-e^4/c^3*b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*B-3*e^4/c^2*b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^
(1/2))*A+5*e^4/c^3*b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B+e^4/c^2*b*(e*x+d)^(1/
2)/(c*e*x+b*e)*A-3*e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d+3*e^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^2+e/b*(e*x+d)^(1/
2)/(c*e*x+b*e)*B*d^3-3*e^2/c*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^2+3*e^2/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*
c/((b*e-c*d)*c)^(1/2))*A*d^2+5*e^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+e/b/(
(b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^3+9*e^2/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d
)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^2-2*c/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^4+4*c^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*d^4+2/3*e^2/c^2*B*(e*x+d)^(3/
2)-2*d^(7/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*B+2*e^3/c^2*A*(e*x+d)^(1/2)+4*d^(7/2)/b^3*arctanh((e*x+d)^(1/2
)/d^(1/2))*A*c-4*e^3/c^3*b*B*(e*x+d)^(1/2)-7*e*d^(5/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*A+6*e^2/c^2*B*d*(e*x
+d)^(1/2)-d^3/b^2*A*(e*x+d)^(1/2)/x-13*e^3/c^2*b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2
))*B*d-e*c/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^3+3*e^3/c^2*b*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d-9*e*c/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

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

[Out]

Exception raised: ValueError

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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)^2,x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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Giac [B]  time = 1.44895, size = 863, normalized size = 2.96 \begin{align*} \frac{{\left (2 \, B b d^{4} - 4 \, A c d^{4} + 7 \, A b d^{3} e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} - \frac{{\left (2 \, B b c^{4} d^{4} - 4 \, A c^{5} d^{4} - B b^{2} c^{3} d^{3} e + 9 \, A b c^{4} d^{3} e - 9 \, B b^{3} c^{2} d^{2} e^{2} - 3 \, A b^{2} c^{3} d^{2} e^{2} + 13 \, B b^{4} c d e^{3} - 5 \, A b^{3} c^{2} d e^{3} - 5 \, B b^{5} e^{4} + 3 \, A b^{4} c e^{4}\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{\sqrt{-c^{2} d + b c e} b^{3} c^{3}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B c^{4} e^{2} + 9 \, \sqrt{x e + d} B c^{4} d e^{2} - 6 \, \sqrt{x e + d} B b c^{3} e^{3} + 3 \, \sqrt{x e + d} A c^{4} e^{3}\right )}}{3 \, c^{6}} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} B b c^{3} d^{3} e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{4} d^{3} e - \sqrt{x e + d} B b c^{3} d^{4} e + 2 \, \sqrt{x e + d} A c^{4} d^{4} e - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} c^{2} d^{2} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} A b c^{3} d^{2} e^{2} + 3 \, \sqrt{x e + d} B b^{2} c^{2} d^{3} e^{2} - 4 \, \sqrt{x e + d} A b c^{3} d^{3} e^{2} + 3 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{3} c d e^{3} - 3 \,{\left (x e + d\right )}^{\frac{3}{2}} A b^{2} c^{2} d e^{3} - 3 \, \sqrt{x e + d} B b^{3} c d^{2} e^{3} + 3 \, \sqrt{x e + d} A b^{2} c^{2} d^{2} e^{3} -{\left (x e + d\right )}^{\frac{3}{2}} B b^{4} e^{4} +{\left (x e + d\right )}^{\frac{3}{2}} A b^{3} c e^{4} + \sqrt{x e + d} B b^{4} d e^{4} - \sqrt{x e + d} A b^{3} c d e^{4}}{{\left ({\left (x e + d\right )}^{2} c - 2 \,{\left (x e + d\right )} c d + c d^{2} +{\left (x e + d\right )} b e - b d e\right )} b^{2} c^{3}} \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)^2,x, algorithm="giac")

[Out]

(2*B*b*d^4 - 4*A*c*d^4 + 7*A*b*d^3*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) - (2*B*b*c^4*d^4 - 4*A*c^5
*d^4 - B*b^2*c^3*d^3*e + 9*A*b*c^4*d^3*e - 9*B*b^3*c^2*d^2*e^2 - 3*A*b^2*c^3*d^2*e^2 + 13*B*b^4*c*d*e^3 - 5*A*
b^3*c^2*d*e^3 - 5*B*b^5*e^4 + 3*A*b^4*c*e^4)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-c^2*d + b*c*e
)*b^3*c^3) + 2/3*((x*e + d)^(3/2)*B*c^4*e^2 + 9*sqrt(x*e + d)*B*c^4*d*e^2 - 6*sqrt(x*e + d)*B*b*c^3*e^3 + 3*sq
rt(x*e + d)*A*c^4*e^3)/c^6 + ((x*e + d)^(3/2)*B*b*c^3*d^3*e - 2*(x*e + d)^(3/2)*A*c^4*d^3*e - sqrt(x*e + d)*B*
b*c^3*d^4*e + 2*sqrt(x*e + d)*A*c^4*d^4*e - 3*(x*e + d)^(3/2)*B*b^2*c^2*d^2*e^2 + 3*(x*e + d)^(3/2)*A*b*c^3*d^
2*e^2 + 3*sqrt(x*e + d)*B*b^2*c^2*d^3*e^2 - 4*sqrt(x*e + d)*A*b*c^3*d^3*e^2 + 3*(x*e + d)^(3/2)*B*b^3*c*d*e^3
- 3*(x*e + d)^(3/2)*A*b^2*c^2*d*e^3 - 3*sqrt(x*e + d)*B*b^3*c*d^2*e^3 + 3*sqrt(x*e + d)*A*b^2*c^2*d^2*e^3 - (x
*e + d)^(3/2)*B*b^4*e^4 + (x*e + d)^(3/2)*A*b^3*c*e^4 + sqrt(x*e + d)*B*b^4*d*e^4 - sqrt(x*e + d)*A*b^3*c*d*e^
4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c^3)