3.1875 \(\int \frac{(A+B x) (d+e x)^{9/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=489 \[ -\frac{21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{105 e^3 (a+b x) \sqrt{d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (a+b x) \sqrt{b d-a e} (-11 a B e+3 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

[Out]

(105*e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*
e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*(d + e*x)^(3/2))/(64*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (21*e^2*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(5/2))/(64*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - (3*e*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(32*b^3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - ((8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(9/2))/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(11/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10
5*e^3*Sqrt[b*d - a*e]*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/(64*b^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.419648, antiderivative size = 489, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac{105 e^3 (a+b x) \sqrt{d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 (a+b x) \sqrt{b d-a e} (-11 a B e+3 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac{3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(105*e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*Sqrt[d + e*x])/(64*b^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*
e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*(d + e*x)^(3/2))/(64*b^5*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
]) - (21*e^2*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(5/2))/(64*b^4*(b*d - a*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]
) - (3*e*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(7/2))/(32*b^3*(b*d - a*e)*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) - ((8*b*B*d + 3*A*b*e - 11*a*B*e)*(d + e*x)^(9/2))/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) - ((A*b - a*B)*(d + e*x)^(11/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (10
5*e^3*Sqrt[b*d - a*e]*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/(64*b^(13/2)*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)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac{(A+B x) (d+e x)^{9/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{9/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (3 e (8 b B d+3 A b e-11 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 )^3} \, dx}{16 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (21 e^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (105 e^3 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac{(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (105 e^3 \left (b^2 d-a b e\right ) (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac{\sqrt{d+e x}}{a b+b^2 x} \, dx}{128 b^6 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt{d+e x}}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (105 e^3 \left (b^2 d-a b e\right )^2 (8 b B d+3 A b e-11 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}{128 b^8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt{d+e x}}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (105 e^2 \left (b^2 d-a b e\right )^2 (8 b B d+3 A b e-11 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 )}{64 b^8 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt{d+e x}}{64 b^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{105 e^3 \sqrt{b d-a e} (8 b B d+3 A b e-11 a B e) (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 b^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.204214, size = 117, normalized size = 0.24 \[ \frac{(d+e x)^{11/2} \left (\frac{e^3 (a+b x)^4 (-11 a B e+3 A b e+8 b B d) \, _2F_1\left (4,\frac{11}{2};\frac{13}{2};\frac{b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+11 (a B-A b)\right )}{44 b (a+b x)^3 \sqrt{(a+b x)^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(9/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((d + e*x)^(11/2)*(11*(-(A*b) + a*B) + (e^3*(8*b*B*d + 3*A*b*e - 11*a*B*e)*(a + b*x)^4*Hypergeometric2F1[4, 11
/2, 13/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^4))/(44*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.034, size = 2430, normalized size = 5. \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)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(20790*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^4*b^2*e^6+6144*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^
(1/2)*x^3*a*b^4*d*e^4+9216*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^3*d*e^4+6144*B*((a*e-b*d)*b)^(1/2)*(e
*x+d)^(1/2)*x*a^3*b^2*d*e^4-5670*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^3*b^3*e^6-1929*A*((a*e-b*
d)*b)^(1/2)*(e*x+d)^(3/2)*b^5*d^3*e+945*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^4*b^2*d*e^5-5025*B*((a
*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^4*b*e^4-5985*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^5*b*d*e^5+2520*B
*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^4*b^2*d^2*e^4+945*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^4*b*e^5
+561*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^5*d^4*e-2295*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a^2*b^3*e^2-945*A*
arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^4*a*b^5*e^6+945*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^
4*b^6*d*e^5+128*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^4*b^5*e^4+3465*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1
/2))*x^4*a^2*b^4*e^6+2520*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^4*b^6*d^2*e^4+384*A*((a*e-b*d)*b)^(1
/2)*(e*x+d)^(1/2)*x^4*b^5*e^5-3780*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a^2*b^4*e^6+13860*B*arcta
n((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a^3*b^3*e^6+975*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a*b^4*e^2-975*A
*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^5*d*e+2295*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^2*b^3*e^3+2295*A*((a*e-b
*d)*b)^(1/2)*(e*x+d)^(5/2)*b^5*d^2*e-3780*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^4*b^2*e^6-5855*B*(
(a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^3*b^2*e^3+13860*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^5*b*e^6+1
929*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^3*b^2*e^4+3465*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^6*e^6
-1320*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*b^5*d^2+3560*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*b^5*d^3-945*A*arcta
n((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*a^5*b*e^6-3224*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*b^5*d^4-3465*B*((a*e
-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^5*e^5+984*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*b^5*d^5+8700*B*((a*e-b*d)*b)^(1/2
)*(e*x+d)^(1/2)*a^4*b*d*e^4+3615*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*a*b^4*d*e+768*B*((a*e-b*d)*b)^(1/2)*(e*x+
d)^(3/2)*x^2*a^2*b^3*e^4-7680*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*a^2*b^3*e^5-35910*B*arctan((e*x+d)^(1/2)
*b/((a*e-b*d)*b)^(1/2))*x^2*a^3*b^3*d*e^5-5985*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^4*a*b^5*d*e^5+3
780*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a*b^5*d*e^5+512*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x^3*
a*b^4*e^4+15120*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^2*b^4*d^2*e^4-4590*A*((a*e-b*d)*b)^(1/2)*(
e*x+d)^(5/2)*a*b^4*d*e^2+2304*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^2*b^3*e^5+3780*A*arctan((e*x+d)^(1/2)*
b/((a*e-b*d)*b)^(1/2))*x*a^3*b^3*d*e^5+15270*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*a^2*b^3*d*e^2-12975*B*((a*e-b
*d)*b)^(1/2)*(e*x+d)^(5/2)*a*b^4*d^2*e+512*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*x*a^3*b^2*e^4-13206*B*((a*e-b*d
)*b)^(1/2)*(e*x+d)^(1/2)*a^3*b^2*d^2*e^3+12084*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^3*d^3*e^2-5481*B*((a*
e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a*b^4*d^4*e-11520*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^2*a^3*b^2*e^5-23940*B*ar
ctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x*a^4*b^2*d*e^5+10080*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*
x*a^3*b^3*d^2*e^4-5787*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^3*d*e^3+5787*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3
/2)*a*b^4*d^2*e^2+1536*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^3*b^2*e^5+18683*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(
3/2)*a^3*b^2*d*e^3-25131*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*a^2*b^3*d^2*e^2+14825*B*((a*e-b*d)*b)^(1/2)*(e*x+
d)^(3/2)*a*b^4*d^3*e-7680*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x*a^4*b*e^5-2244*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(
1/2)*a^3*b^2*d*e^4+3366*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*a^2*b^3*d^2*e^3-2244*A*((a*e-b*d)*b)^(1/2)*(e*x+d)
^(1/2)*a*b^4*d^3*e^2-1920*B*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^4*a*b^4*e^5+1536*B*((a*e-b*d)*b)^(1/2)*(e*x+d)
^(1/2)*x^4*b^5*d*e^4-23940*B*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a^2*b^4*d*e^5+10080*B*arctan((e*x
+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^3*a*b^5*d^2*e^4+1536*A*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*x^3*a*b^4*e^5+5670
*A*arctan((e*x+d)^(1/2)*b/((a*e-b*d)*b)^(1/2))*x^2*a^2*b^4*d*e^5)/e*(b*x+a)/((a*e-b*d)*b)^(1/2)/b^6/((b*x+a)^2
)^(5/2)

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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{9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(9/2)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2), x)

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Fricas [A]  time = 1.78657, size = 3054, normalized size = 6.25 \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)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/384*(315*(8*B*a^4*b*d*e^3 - (11*B*a^5 - 3*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4
+ 4*(8*B*a*b^4*d*e^3 - (11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (11*B*a^3*b^2 - 3*A*a^2*b^
3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (11*B*a^4*b - 3*A*a^3*b^2)*e^4)*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*(128*B*b^5*e^4*x^5 - 16*(B*a*b^4 + 3*A*b^5)*d^4
- 72*(B*a^2*b^3 + A*a*b^4)*d^3*e - 126*(3*B*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 105*(35*B*a^4*b - 3*A*a^3*b^2)*d*e
^3 - 315*(11*B*a^5 - 3*A*a^4*b)*e^4 + 128*(13*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 - (1320*B*b^5*d^2*
e^2 - (10271*B*a*b^4 - 975*A*b^5)*d*e^3 + 837*(11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 - (400*B*b^5*d^3*e + 30*(71*
B*a*b^4 + 21*A*b^5)*d^2*e^2 - 9*(2041*B*a^2*b^3 - 185*A*a*b^4)*d*e^3 + 1533*(11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*
x^2 - (64*B*b^5*d^4 + 8*(35*B*a*b^4 + 33*A*b^5)*d^3*e + 36*(41*B*a^2*b^3 + 13*A*a*b^4)*d^2*e^2 - 21*(649*B*a^3
*b^2 - 57*A*a^2*b^3)*d*e^3 + 1155*(11*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^10*x^4 + 4*a*b^9*x^3 +
6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6), -1/192*(315*(8*B*a^4*b*d*e^3 - (11*B*a^5 - 3*A*a^4*b)*e^4 + (8*B*b^5*d
*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2
*b^3*d*e^3 - (11*B*a^3*b^2 - 3*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (11*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)
*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (128*B*b^5*e^4*x^5 - 16*(B*a
*b^4 + 3*A*b^5)*d^4 - 72*(B*a^2*b^3 + A*a*b^4)*d^3*e - 126*(3*B*a^3*b^2 + A*a^2*b^3)*d^2*e^2 + 105*(35*B*a^4*b
- 3*A*a^3*b^2)*d*e^3 - 315*(11*B*a^5 - 3*A*a^4*b)*e^4 + 128*(13*B*b^5*d*e^3 - (11*B*a*b^4 - 3*A*b^5)*e^4)*x^4
- (1320*B*b^5*d^2*e^2 - (10271*B*a*b^4 - 975*A*b^5)*d*e^3 + 837*(11*B*a^2*b^3 - 3*A*a*b^4)*e^4)*x^3 - (400*B*
b^5*d^3*e + 30*(71*B*a*b^4 + 21*A*b^5)*d^2*e^2 - 9*(2041*B*a^2*b^3 - 185*A*a*b^4)*d*e^3 + 1533*(11*B*a^3*b^2 -
3*A*a^2*b^3)*e^4)*x^2 - (64*B*b^5*d^4 + 8*(35*B*a*b^4 + 33*A*b^5)*d^3*e + 36*(41*B*a^2*b^3 + 13*A*a*b^4)*d^2*
e^2 - 21*(649*B*a^3*b^2 - 57*A*a^2*b^3)*d*e^3 + 1155*(11*B*a^4*b - 3*A*a^3*b^2)*e^4)*x)*sqrt(e*x + d))/(b^10*x
^4 + 4*a*b^9*x^3 + 6*a^2*b^8*x^2 + 4*a^3*b^7*x + a^4*b^6)]

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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)**(9/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.43154, size = 1177, normalized size = 2.41 \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)^(9/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

105/64*(8*B*b^2*d^2*e^3 - 19*B*a*b*d*e^4 + 3*A*b^2*d*e^4 + 11*B*a^2*e^5 - 3*A*a*b*e^5)*arctan(sqrt(x*e + d)*b/
sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^6*sgn((x*e + d)*b*e - b*d*e + a*e^2)) - 1/192*(1320*(x*e + d)^(7
/2)*B*b^5*d^2*e^3 - 3560*(x*e + d)^(5/2)*B*b^5*d^3*e^3 + 3224*(x*e + d)^(3/2)*B*b^5*d^4*e^3 - 984*sqrt(x*e + d
)*B*b^5*d^5*e^3 - 3615*(x*e + d)^(7/2)*B*a*b^4*d*e^4 + 975*(x*e + d)^(7/2)*A*b^5*d*e^4 + 12975*(x*e + d)^(5/2)
*B*a*b^4*d^2*e^4 - 2295*(x*e + d)^(5/2)*A*b^5*d^2*e^4 - 14825*(x*e + d)^(3/2)*B*a*b^4*d^3*e^4 + 1929*(x*e + d)
^(3/2)*A*b^5*d^3*e^4 + 5481*sqrt(x*e + d)*B*a*b^4*d^4*e^4 - 561*sqrt(x*e + d)*A*b^5*d^4*e^4 + 2295*(x*e + d)^(
7/2)*B*a^2*b^3*e^5 - 975*(x*e + d)^(7/2)*A*a*b^4*e^5 - 15270*(x*e + d)^(5/2)*B*a^2*b^3*d*e^5 + 4590*(x*e + d)^
(5/2)*A*a*b^4*d*e^5 + 25131*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^5 - 5787*(x*e + d)^(3/2)*A*a*b^4*d^2*e^5 - 12084*s
qrt(x*e + d)*B*a^2*b^3*d^3*e^5 + 2244*sqrt(x*e + d)*A*a*b^4*d^3*e^5 + 5855*(x*e + d)^(5/2)*B*a^3*b^2*e^6 - 229
5*(x*e + d)^(5/2)*A*a^2*b^3*e^6 - 18683*(x*e + d)^(3/2)*B*a^3*b^2*d*e^6 + 5787*(x*e + d)^(3/2)*A*a^2*b^3*d*e^6
+ 13206*sqrt(x*e + d)*B*a^3*b^2*d^2*e^6 - 3366*sqrt(x*e + d)*A*a^2*b^3*d^2*e^6 + 5153*(x*e + d)^(3/2)*B*a^4*b
*e^7 - 1929*(x*e + d)^(3/2)*A*a^3*b^2*e^7 - 7164*sqrt(x*e + d)*B*a^4*b*d*e^7 + 2244*sqrt(x*e + d)*A*a^3*b^2*d*
e^7 + 1545*sqrt(x*e + d)*B*a^5*e^8 - 561*sqrt(x*e + d)*A*a^4*b*e^8)/(((x*e + d)*b - b*d + a*e)^4*b^6*sgn((x*e
+ d)*b*e - b*d*e + a*e^2)) + 2/3*((x*e + d)^(3/2)*B*b^10*e^3 + 12*sqrt(x*e + d)*B*b^10*d*e^3 - 15*sqrt(x*e + d
)*B*a*b^9*e^4 + 3*sqrt(x*e + d)*A*b^10*e^4)/(b^15*sgn((x*e + d)*b*e - b*d*e + a*e^2))