3.1229 \(\int \frac{(A+B x) (d+e x)^{7/2}}{b x+c x^2} \, dx\)

Optimal. Leaf size=228 \[ \frac{2 \sqrt{d+e x} \left (A c e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )+B (c d-b e)^3\right )}{c^4}+\frac{2 (d+e x)^{5/2} (A c e-b B e+B c d)}{5 c^2}+\frac{2 (d+e x)^{3/2} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{3 c^3}-\frac{2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{9/2}}-\frac{2 A d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B (d+e x)^{7/2}}{7 c} \]

[Out]

(2*(B*(c*d - b*e)^3 + A*c*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2))*Sqrt[d + e*x])/c^4 + (2*(B*(c*d - b*e)^2 + A*c*
e*(2*c*d - b*e))*(d + e*x)^(3/2))/(3*c^3) + (2*(B*c*d - b*B*e + A*c*e)*(d + e*x)^(5/2))/(5*c^2) + (2*B*(d + e*
x)^(7/2))/(7*c) - (2*A*d^(7/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b - (2*(b*B - A*c)*(c*d - b*e)^(7/2)*ArcTanh[(S
qrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*c^(9/2))

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Rubi [A]  time = 0.560963, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {824, 826, 1166, 208} \[ \frac{2 \sqrt{d+e x} \left (A c e \left (b^2 e^2-3 b c d e+3 c^2 d^2\right )+B (c d-b e)^3\right )}{c^4}+\frac{2 (d+e x)^{5/2} (A c e-b B e+B c d)}{5 c^2}+\frac{2 (d+e x)^{3/2} \left (A c e (2 c d-b e)+B (c d-b e)^2\right )}{3 c^3}-\frac{2 (b B-A c) (c d-b e)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b c^{9/2}}-\frac{2 A d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b}+\frac{2 B (d+e x)^{7/2}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(B*(c*d - b*e)^3 + A*c*e*(3*c^2*d^2 - 3*b*c*d*e + b^2*e^2))*Sqrt[d + e*x])/c^4 + (2*(B*(c*d - b*e)^2 + A*c*
e*(2*c*d - b*e))*(d + e*x)^(3/2))/(3*c^3) + (2*(B*c*d - b*B*e + A*c*e)*(d + e*x)^(5/2))/(5*c^2) + (2*B*(d + e*
x)^(7/2))/(7*c) - (2*A*d^(7/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b - (2*(b*B - A*c)*(c*d - b*e)^(7/2)*ArcTanh[(S
qrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*c^(9/2))

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

Mathematica [A]  time = 0.517364, size = 212, normalized size = 0.93 \[ \frac{2 \left (\frac{(b B-A c) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt{c} \sqrt{d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )}{c^{9/2}}+A \sqrt{d+e x} \left (122 d^2 e x+176 d^3+66 d e^2 x^2+15 e^3 x^3\right )-105 A d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right )}{105 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(A*Sqrt[d + e*x]*(176*d^3 + 122*d^2*e*x + 66*d*e^2*x^2 + 15*e^3*x^3) - 105*A*d^(7/2)*ArcTanh[Sqrt[d + e*x]/
Sqrt[d]] + ((b*B - A*c)*(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^(9/2)))/(105*b)

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Maple [B]  time = 0.02, size = 741, 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]

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

[Out]

-8/c*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*e+8/c^2*b^2/((b*e-c*d)*c)^(1/2)*a
rctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d*e^3-12/c*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*e^2-8/c^3*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*e^3+2/7*B*
(e*x+d)^(7/2)/c+2/c^4*b^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*e^4-2*c/b/((b*e-c*
d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^4-6/c^2*B*b*d^2*e*(e*x+d)^(1/2)-2/c^3*b^3/((b*e-c*
d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*e^4-4/3/c^2*B*(e*x+d)^(3/2)*b*d*e-6/c^2*A*b*d*e^2*(e
*x+d)^(1/2)+6/c^3*B*b^2*d*e^2*(e*x+d)^(1/2)+2/3/c*B*(e*x+d)^(3/2)*d^2+2/c*B*d^3*(e*x+d)^(1/2)+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+2/5/c*A*(e*x+d)^(5/2)*e+2/5/c*B*(e*x+d)^(5/2)*d-2*A*d^
(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b+2/3/c^3*B*(e*x+d)^(3/2)*b^2*e^2+2/c^3*A*b^2*e^3*(e*x+d)^(1/2)+6/c*A*d^2
*e*(e*x+d)^(1/2)-2/c^4*B*e^3*b^3*(e*x+d)^(1/2)+8/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2
))*A*d^3*e-2/5/c^2*B*(e*x+d)^(5/2)*b*e-2/3/c^2*A*(e*x+d)^(3/2)*b*e^2+4/3/c*A*(e*x+d)^(3/2)*d*e+12/c^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*e^2

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 93.0426, size = 3148, normalized size = 13.81 \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),x, algorithm="fricas")

[Out]

[1/105*(105*A*c^4*d^(7/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 105*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2
*c^2 - A*b*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A*b^3*c)*e^3)*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*(15*B*b*c^3*e^3*x^3 + 176*B*b*c^3*d^3 -
406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c - A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*
d*e^2 - 7*(B*b^2*c^2 - A*b*c^3)*e^3)*x^2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(B*b^3*c
- A*b^2*c^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), 1/105*(105*A*c^4*d^(7/2)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d
)/x) - 210*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A
*b^3*c)*e^3)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + 2*(15*B*b*c^3*e^
3*x^3 + 176*B*b*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c - A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b
^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 - A*b*c^3)*e^3)*x^2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b
*c^3)*d*e^2 + 35*(B*b^3*c - A*b^2*c^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), 1/105*(210*A*c^4*sqrt(-d)*d^3*arctan(sq
rt(e*x + d)*sqrt(-d)/d) - 105*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)
*d*e^2 - (B*b^4 - A*b^3*c)*e^3)*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*(15*B*b*c^3*e^3*x^3 + 176*B*b*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*e + 350*(B*b^3*c
- A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 - A*b*c^3)*e^3)*x^2 + (122
*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(B*b^3*c - A*b^2*c^2)*e^3)*x)*sqrt(e*x + d))/(b*c^4), 2/
105*(105*A*c^4*sqrt(-d)*d^3*arctan(sqrt(e*x + d)*sqrt(-d)/d) - 105*((B*b*c^3 - A*c^4)*d^3 - 3*(B*b^2*c^2 - A*b
*c^3)*d^2*e + 3*(B*b^3*c - A*b^2*c^2)*d*e^2 - (B*b^4 - A*b^3*c)*e^3)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d
)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*e)) + (15*B*b*c^3*e^3*x^3 + 176*B*b*c^3*d^3 - 406*(B*b^2*c^2 - A*b*c^3)*d^2*
e + 350*(B*b^3*c - A*b^2*c^2)*d*e^2 - 105*(B*b^4 - A*b^3*c)*e^3 + 3*(22*B*b*c^3*d*e^2 - 7*(B*b^2*c^2 - A*b*c^3
)*e^3)*x^2 + (122*B*b*c^3*d^2*e - 112*(B*b^2*c^2 - A*b*c^3)*d*e^2 + 35*(B*b^3*c - A*b^2*c^2)*e^3)*x)*sqrt(e*x
+ d))/(b*c^4)]

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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),x)

[Out]

Timed out

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Giac [B]  time = 1.31357, size = 643, normalized size = 2.82 \begin{align*} \frac{2 \, A d^{4} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b \sqrt{-d}} + \frac{2 \,{\left (B b c^{4} d^{4} - A c^{5} d^{4} - 4 \, B b^{2} c^{3} d^{3} e + 4 \, A b c^{4} d^{3} e + 6 \, B b^{3} c^{2} d^{2} e^{2} - 6 \, A b^{2} c^{3} d^{2} e^{2} - 4 \, B b^{4} c d e^{3} + 4 \, A b^{3} c^{2} d e^{3} + B b^{5} e^{4} - 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 c^{4}} + \frac{2 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} B c^{6} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B c^{6} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B c^{6} d^{2} + 105 \, \sqrt{x e + d} B c^{6} d^{3} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} B b c^{5} e + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} A c^{6} e - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} B b c^{5} d e + 70 \,{\left (x e + d\right )}^{\frac{3}{2}} A c^{6} d e - 315 \, \sqrt{x e + d} B b c^{5} d^{2} e + 315 \, \sqrt{x e + d} A c^{6} d^{2} e + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} B b^{2} c^{4} e^{2} - 35 \,{\left (x e + d\right )}^{\frac{3}{2}} A b c^{5} e^{2} + 315 \, \sqrt{x e + d} B b^{2} c^{4} d e^{2} - 315 \, \sqrt{x e + d} A b c^{5} d e^{2} - 105 \, \sqrt{x e + d} B b^{3} c^{3} e^{3} + 105 \, \sqrt{x e + d} A b^{2} c^{4} e^{3}\right )}}{105 \, c^{7}} \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),x, algorithm="giac")

[Out]

2*A*d^4*arctan(sqrt(x*e + d)/sqrt(-d))/(b*sqrt(-d)) + 2*(B*b*c^4*d^4 - A*c^5*d^4 - 4*B*b^2*c^3*d^3*e + 4*A*b*c
^4*d^3*e + 6*B*b^3*c^2*d^2*e^2 - 6*A*b^2*c^3*d^2*e^2 - 4*B*b^4*c*d*e^3 + 4*A*b^3*c^2*d*e^3 + B*b^5*e^4 - 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*c^4) + 2/105*(15*(x*e + d)^(7/2)*
B*c^6 + 21*(x*e + d)^(5/2)*B*c^6*d + 35*(x*e + d)^(3/2)*B*c^6*d^2 + 105*sqrt(x*e + d)*B*c^6*d^3 - 21*(x*e + d)
^(5/2)*B*b*c^5*e + 21*(x*e + d)^(5/2)*A*c^6*e - 70*(x*e + d)^(3/2)*B*b*c^5*d*e + 70*(x*e + d)^(3/2)*A*c^6*d*e
- 315*sqrt(x*e + d)*B*b*c^5*d^2*e + 315*sqrt(x*e + d)*A*c^6*d^2*e + 35*(x*e + d)^(3/2)*B*b^2*c^4*e^2 - 35*(x*e
+ d)^(3/2)*A*b*c^5*e^2 + 315*sqrt(x*e + d)*B*b^2*c^4*d*e^2 - 315*sqrt(x*e + d)*A*b*c^5*d*e^2 - 105*sqrt(x*e +
d)*B*b^3*c^3*e^3 + 105*sqrt(x*e + d)*A*b^2*c^4*e^3)/c^7