3.1866 \(\int \frac{(A+B x) (d+e x)^{7/2}}{(a^2+2 a b x+b^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=407 \[ -\frac{(d+e x)^{9/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{7 e (a+b x) (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{7 e (a+b x) (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) \sqrt{d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (a+b x) (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*(d + e*x)^(3/2))/(12*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*
(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
((4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*
B)*(d + e*x)^(9/2))/(2*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*
d + 5*A*b*e - 9*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(11/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])

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Rubi [A]  time = 0.376578, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {770, 78, 47, 50, 63, 208} \[ -\frac{(d+e x)^{9/2} (A b-a B)}{2 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{7/2} (-9 a B e+5 A b e+4 b B d)}{4 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{7 e (a+b x) (d+e x)^{5/2} (-9 a B e+5 A b e+4 b B d)}{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{7 e (a+b x) (d+e x)^{3/2} (-9 a B e+5 A b e+4 b B d)}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (a+b x) \sqrt{d+e x} (b d-a e) (-9 a B e+5 A b e+4 b B d)}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (a+b x) (b d-a e)^{3/2} (-9 a B e+5 A b e+4 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*e*(b*d - a*e)*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(4*b^5*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (7*e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*(d + e*x)^(3/2))/(12*b^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (7*e*
(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)*(d + e*x)^(5/2))/(20*b^3*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) -
((4*b*B*d + 5*A*b*e - 9*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((A*b - a*
B)*(d + e*x)^(9/2))/(2*b*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (7*e*(b*d - a*e)^(3/2)*(4*b*B*
d + 5*A*b*e - 9*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*b^(11/2)*Sqrt[a^2 + 2*a*
b*x + b^2*x^2])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac{(A+B x) (d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left ((4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 e (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{5/2}}{a b+b^2 x} \, dx}{8 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 e \left (b^2 d-a b e\right ) (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{3/2}}{a b+b^2 x} \, dx}{8 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 e \left (b^2 d-a b e\right )^2 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{a b+b^2 x} \, dx}{8 b^6 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt{d+e x}}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 e \left (b^2 d-a b e\right )^3 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \int \frac{1}{\left (a b+b^2 x\right ) \sqrt{d+e x}} \, dx}{8 b^8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt{d+e x}}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 \left (b^2 d-a b e\right )^3 (4 b B d+5 A b e-9 a B e) \left (a b+b^2 x\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a b-\frac{b^2 d}{e}+\frac{b^2 x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 b^8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{7 e (b d-a e) (4 b B d+5 A b e-9 a B e) (a+b x) \sqrt{d+e x}}{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{3/2}}{12 b^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 e (4 b B d+5 A b e-9 a B e) (a+b x) (d+e x)^{5/2}}{20 b^3 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(4 b B d+5 A b e-9 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{9/2}}{2 b (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{7 e (b d-a e)^{3/2} (4 b B d+5 A b e-9 a B e) (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 b^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.137131, size = 111, normalized size = 0.27 \[ \frac{(a+b x) (d+e x)^{9/2} \left (\frac{e (a+b x)^2 (-9 a B e+5 A b e+4 b B d) \, _2F_1\left (2,\frac{9}{2};\frac{11}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^2}+9 a B-9 A b\right )}{18 b \left ((a+b x)^2\right )^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*(d + e*x)^(9/2)*(-9*A*b + 9*a*B + (e*(4*b*B*d + 5*A*b*e - 9*a*B*e)*(a + b*x)^2*Hypergeometric2F1[2,
9/2, 11/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^2))/(18*b*(b*d - a*e)*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.027, size = 1873, normalized size = 4.6 \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)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

1/60*(-525*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b*e^4+80*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a*b^3*e^3-36
0*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a*b^3*e^4-1890*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^4*b
*e^5-155*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^2*e^3-195*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^4*d^2*e-105
0*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^3*b^2*d*e^4+525*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2)
)*a^2*b^3*d^2*e^3+135*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^3*b*e^3+2310*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b
)^(1/2))*a^4*b*d*e^4-1785*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^3*b^2*d^2*e^3+420*B*arctan((e*x+d)^(
1/2)*b/((a*e-b*d)*b)^(1/2))*a^2*b^3*d^3*e^2+40*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*b^4*e^3+24*B*((a*e-b*d)
*b)^(1/2)*(e*x+d)^(5/2)*x^2*b^4*e^2+24*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^2*b^2*e^2+420*B*arctan((e*x+d)^(1
/2)*b/((a*e-b*d)*b)^(1/2))*x^2*b^5*d^3*e^2+1050*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^3*b^2*e^5+16
5*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^4*d^3*e+525*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^2*b^3*
e^5+525*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*b^5*d^2*e^3-945*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*
b)^(1/2))*x^2*a^3*b^2*e^5+720*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a*b^3*d^2*e^2-945*B*arctan((e*x+d)^(1/2)*b
/((a*e-b*d)*b)^(1/2))*a^5*e^5+945*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^4*e^4+60*B*((a*e-b*d)*b)^(1/2)*(e*x+d)
^(1/2)*b^4*d^4+525*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^4*b*e^5-60*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3
/2)*b^4*d^3+390*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^3*d*e^2-720*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^2*
b^2*e^4-490*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^2*d*e^2+80*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*b^4*d
*e^2-1050*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a*b^4*d*e^4+48*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)
*x*a*b^3*e^2-120*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^2*a*b^3*e^3-1815*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^
3*b*d*e^3+2310*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^2*b^3*d*e^4-1785*B*arctan((e*x+d)^(1/2)*b/(
(a*e-b*d)*b)^(1/2))*x^2*a*b^4*d^2*e^3+840*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a*b^4*d^3*e^2+1215*B
*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d^2*e^2-405*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^3*e+375*B*(
(a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a*b^3*d^2*e+1440*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^3*b*e^4+855*A*((a*e-
b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^2*d*e^3-495*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^3*d^2*e^2-1080*B*((a*e-b
*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a*b^3*d*e^3+720*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a*b^3*d*e^3-2160*B*((a*e-
b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^2*b^2*d*e^3+160*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a*b^3*d*e^2+360*A*((a*e-
b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*b^4*d*e^3-2100*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^2*b^3*d*e^4+1
050*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a*b^4*d^2*e^3-240*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a^
2*b^2*e^3+720*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^2*e^4+360*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*
b^4*d^2*e^2+4620*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^3*b^2*d*e^4-3570*B*arctan((e*x+d)^(1/2)*b/(
(a*e-b*d)*b)^(1/2))*x*a^2*b^3*d^2*e^3)/e*(b*x+a)/((a*e-b*d)*b)^(1/2)/b^5/((b*x+a)^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(b^2*x^2 + 2*a*b*x + a^2)^(3/2), x)

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Fricas [A]  time = 1.7524, size = 2276, normalized size = 5.59 \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)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

[1/120*(105*(4*B*a^2*b^2*d^2*e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5*A*a^3*b)*e^3 + (4*B*b^4*d^2*e
- (13*B*a*b^3 - 5*A*b^4)*d*e^2 + (9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + 2*(4*B*a*b^3*d^2*e - (13*B*a^2*b^2 - 5*
A*a*b^3)*d*e^2 + (9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x +
d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) + 2*(24*B*b^4*e^3*x^4 - 30*(B*a*b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15
*A*a*b^3)*d^2*e - 140*(12*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e^2 - (
9*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e - 2*(59*B*a*b^3 - 25*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 5*A*a*b
^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3 - 195*A*b^4)*d^2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*
(9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^7*x^2 + 2*a*b^6*x + a^2*b^5), -1/60*(105*(4*B*a^2*b^2*d^2*
e - (13*B*a^3*b - 5*A*a^2*b^2)*d*e^2 + (9*B*a^4 - 5*A*a^3*b)*e^3 + (4*B*b^4*d^2*e - (13*B*a*b^3 - 5*A*b^4)*d*e
^2 + (9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 + 2*(4*B*a*b^3*d^2*e - (13*B*a^2*b^2 - 5*A*a*b^3)*d*e^2 + (9*B*a^3*b -
5*A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (24*B*b
^4*e^3*x^4 - 30*(B*a*b^3 + A*b^4)*d^3 + 7*(107*B*a^2*b^2 - 15*A*a*b^3)*d^2*e - 140*(12*B*a^3*b - 5*A*a^2*b^2)*
d*e^2 + 105*(9*B*a^4 - 5*A*a^3*b)*e^3 + 8*(16*B*b^4*d*e^2 - (9*B*a*b^3 - 5*A*b^4)*e^3)*x^3 + 8*(58*B*b^4*d^2*e
- 2*(59*B*a*b^3 - 25*A*b^4)*d*e^2 + 7*(9*B*a^2*b^2 - 5*A*a*b^3)*e^3)*x^2 - (60*B*b^4*d^3 - (1303*B*a*b^3 - 19
5*A*b^4)*d^2*e + 14*(203*B*a^2*b^2 - 85*A*a*b^3)*d*e^2 - 175*(9*B*a^3*b - 5*A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/
(b^7*x^2 + 2*a*b^6*x + a^2*b^5)]

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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)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.31789, size = 925, normalized size = 2.27 \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)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

7/4*(4*B*b^3*d^3*e^2 - 17*B*a*b^2*d^2*e^3 + 5*A*b^3*d^2*e^3 + 22*B*a^2*b*d*e^4 - 10*A*a*b^2*d*e^4 - 9*B*a^3*e^
5 + 5*A*a^2*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^(-1)/(sqrt(-b^2*d + a*b*e)*b^5*sgn((x*e + d)
*b*e - b*d*e + a*e^2)) - 1/4*(4*(x*e + d)^(3/2)*B*b^4*d^3*e^2 - 4*sqrt(x*e + d)*B*b^4*d^4*e^2 - 25*(x*e + d)^(
3/2)*B*a*b^3*d^2*e^3 + 13*(x*e + d)^(3/2)*A*b^4*d^2*e^3 + 27*sqrt(x*e + d)*B*a*b^3*d^3*e^3 - 11*sqrt(x*e + d)*
A*b^4*d^3*e^3 + 38*(x*e + d)^(3/2)*B*a^2*b^2*d*e^4 - 26*(x*e + d)^(3/2)*A*a*b^3*d*e^4 - 57*sqrt(x*e + d)*B*a^2
*b^2*d^2*e^4 + 33*sqrt(x*e + d)*A*a*b^3*d^2*e^4 - 17*(x*e + d)^(3/2)*B*a^3*b*e^5 + 13*(x*e + d)^(3/2)*A*a^2*b^
2*e^5 + 49*sqrt(x*e + d)*B*a^3*b*d*e^5 - 33*sqrt(x*e + d)*A*a^2*b^2*d*e^5 - 15*sqrt(x*e + d)*B*a^4*e^6 + 11*sq
rt(x*e + d)*A*a^3*b*e^6)*e^(-1)/(((x*e + d)*b - b*d + a*e)^2*b^5*sgn((x*e + d)*b*e - b*d*e + a*e^2)) + 2/15*(3
*(x*e + d)^(5/2)*B*b^12*e^6 + 10*(x*e + d)^(3/2)*B*b^12*d*e^6 + 45*sqrt(x*e + d)*B*b^12*d^2*e^6 - 15*(x*e + d)
^(3/2)*B*a*b^11*e^7 + 5*(x*e + d)^(3/2)*A*b^12*e^7 - 135*sqrt(x*e + d)*B*a*b^11*d*e^7 + 45*sqrt(x*e + d)*A*b^1
2*d*e^7 + 90*sqrt(x*e + d)*B*a^2*b^10*e^8 - 45*sqrt(x*e + d)*A*a*b^11*e^8)*e^(-5)/(b^15*sgn((x*e + d)*b*e - b*
d*e + a*e^2))