3.1879 \(\int \frac {(A+B x) \sqrt {d+e x}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ \frac {e^2 \sqrt {d+e x} (-3 a B e-5 A b e+8 b B d)}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {e \sqrt {d+e x} (-3 a B e-5 A b e+8 b B d)}{96 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {(d+e x)^{3/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {\sqrt {d+e x} (-3 a B e-5 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {e^3 (a+b x) (-3 a B e-5 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}} \]

[Out]

-1/4*(A*b-B*a)*(e*x+d)^(3/2)/b/(-a*e+b*d)/(b*x+a)^3/((b*x+a)^2)^(1/2)-1/64*e^3*(-5*A*b*e-3*B*a*e+8*B*b*d)*(b*x
+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/b^(5/2)/(-a*e+b*d)^(7/2)/((b*x+a)^2)^(1/2)+1/64*e^2*(-5*A*
b*e-3*B*a*e+8*B*b*d)*(e*x+d)^(1/2)/b^2/(-a*e+b*d)^3/((b*x+a)^2)^(1/2)-1/24*(-5*A*b*e-3*B*a*e+8*B*b*d)*(e*x+d)^
(1/2)/b^2/(-a*e+b*d)/(b*x+a)^2/((b*x+a)^2)^(1/2)-1/96*e*(-5*A*b*e-3*B*a*e+8*B*b*d)*(e*x+d)^(1/2)/b^2/(-a*e+b*d
)^2/(b*x+a)/((b*x+a)^2)^(1/2)

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Rubi [A]  time = 0.35, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {770, 78, 47, 51, 63, 208} \[ \frac {e^2 \sqrt {d+e x} (-3 a B e-5 A b e+8 b B d)}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}-\frac {e^3 (a+b x) (-3 a B e-5 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{7/2}}-\frac {e \sqrt {d+e x} (-3 a B e-5 A b e+8 b B d)}{96 b^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {(d+e x)^{3/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {\sqrt {d+e x} (-3 a B e-5 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*(8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(64*b^2*(b*d - a*e)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((8*b
*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(24*b^2*(b*d - a*e)*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(
8*b*B*d - 5*A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(96*b^2*(b*d - a*e)^2*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
(A*b - a*B)*(d + e*x)^(3/2))/(4*b*(b*d - a*e)*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e^3*(8*b*B*d - 5*A
*b*e - 3*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(5/2)*(b*d - a*e)^(7/2)*Sqrt
[a^2 + 2*a*b*x + b^2*x^2])

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 51

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

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]

Rubi steps

\begin {align*} \int \frac {(A+B x) \sqrt {d+e x}}{\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) \sqrt {d+e x}}{\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)^{3/2}}{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-5 A b e-3 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\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-5 A b e-3 a B e) \sqrt {d+e x}}{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)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (e (8 b B d-5 A b e-3 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{48 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{96 b^2 (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (e^2 (8 b B d-5 A b e-3 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{64 b (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {e^2 (8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{64 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{96 b^2 (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (e^3 (8 b B d-5 A b e-3 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^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {e^2 (8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{64 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{96 b^2 (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (e^2 (8 b B d-5 A b e-3 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^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {e^2 (8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{64 b^2 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (8 b B d-5 A b e-3 a B e) \sqrt {d+e x}}{96 b^2 (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{3/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (8 b B d-5 A b e-3 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{5/2} (b d-a e)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 116, normalized size = 0.32 \[ \frac {(d+e x)^{3/2} \left (-\frac {e^3 (a+b x)^4 (3 a B e+5 A b e-8 b B d) \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+3 a B-3 A b\right )}{12 b (a+b x)^3 \sqrt {(a+b x)^2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(3/2)*(-3*A*b + 3*a*B - (e^3*(-8*b*B*d + 5*A*b*e + 3*a*B*e)*(a + b*x)^4*Hypergeometric2F1[3/2, 4, 5
/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^4))/(12*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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fricas [B]  time = 0.72, size = 1841, normalized size = 5.13 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

[1/384*(3*(8*B*a^4*b*d*e^3 - (3*B*a^5 + 5*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (3*B*a*b^4 + 5*A*b^5)*e^4)*x^4 + 4*(
8*B*a*b^4*d*e^3 - (3*B*a^2*b^3 + 5*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (3*B*a^3*b^2 + 5*A*a^2*b^3)*e^4)
*x^2 + 4*(8*B*a^3*b^2*d*e^3 - (3*B*a^4*b + 5*A*a^3*b^2)*e^4)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e -
2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(16*(B*a*b^5 + 3*A*b^6)*d^4 - 8*(7*B*a^2*b^4 + 23*A*a*b^5
)*d^3*e + 2*(29*B*a^3*b^3 + 127*A*a^2*b^4)*d^2*e^2 - (27*B*a^4*b^2 + 133*A*a^3*b^3)*d*e^3 + 3*(3*B*a^5*b + 5*A
*a^4*b^2)*e^4 - 3*(8*B*b^6*d^2*e^2 - (11*B*a*b^5 + 5*A*b^6)*d*e^3 + (3*B*a^2*b^4 + 5*A*a*b^5)*e^4)*x^3 + (16*B
*b^6*d^3*e - 10*(11*B*a*b^5 + A*b^6)*d^2*e^2 + (127*B*a^2*b^4 + 65*A*a*b^5)*d*e^3 - 11*(3*B*a^3*b^3 + 5*A*a^2*
b^4)*e^4)*x^2 + (64*B*b^6*d^4 - 8*(29*B*a*b^5 - A*b^6)*d^3*e + 4*(65*B*a^2*b^4 - 11*A*a*b^5)*d^2*e^2 - (125*B*
a^3*b^3 - 109*A*a^2*b^4)*d*e^3 + (33*B*a^4*b^2 - 73*A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d^4 - 4*a^5*b^6
*d^3*e + 6*a^6*b^5*d^2*e^2 - 4*a^7*b^4*d*e^3 + a^8*b^3*e^4 + (b^11*d^4 - 4*a*b^10*d^3*e + 6*a^2*b^9*d^2*e^2 -
4*a^3*b^8*d*e^3 + a^4*b^7*e^4)*x^4 + 4*(a*b^10*d^4 - 4*a^2*b^9*d^3*e + 6*a^3*b^8*d^2*e^2 - 4*a^4*b^7*d*e^3 + a
^5*b^6*e^4)*x^3 + 6*(a^2*b^9*d^4 - 4*a^3*b^8*d^3*e + 6*a^4*b^7*d^2*e^2 - 4*a^5*b^6*d*e^3 + a^6*b^5*e^4)*x^2 +
4*(a^3*b^8*d^4 - 4*a^4*b^7*d^3*e + 6*a^5*b^6*d^2*e^2 - 4*a^6*b^5*d*e^3 + a^7*b^4*e^4)*x), 1/192*(3*(8*B*a^4*b*
d*e^3 - (3*B*a^5 + 5*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (3*B*a*b^4 + 5*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3 - (3*
B*a^2*b^3 + 5*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (3*B*a^3*b^2 + 5*A*a^2*b^3)*e^4)*x^2 + 4*(8*B*a^3*b^2
*d*e^3 - (3*B*a^4*b + 5*A*a^3*b^2)*e^4)*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e
*x + b*d)) - (16*(B*a*b^5 + 3*A*b^6)*d^4 - 8*(7*B*a^2*b^4 + 23*A*a*b^5)*d^3*e + 2*(29*B*a^3*b^3 + 127*A*a^2*b^
4)*d^2*e^2 - (27*B*a^4*b^2 + 133*A*a^3*b^3)*d*e^3 + 3*(3*B*a^5*b + 5*A*a^4*b^2)*e^4 - 3*(8*B*b^6*d^2*e^2 - (11
*B*a*b^5 + 5*A*b^6)*d*e^3 + (3*B*a^2*b^4 + 5*A*a*b^5)*e^4)*x^3 + (16*B*b^6*d^3*e - 10*(11*B*a*b^5 + A*b^6)*d^2
*e^2 + (127*B*a^2*b^4 + 65*A*a*b^5)*d*e^3 - 11*(3*B*a^3*b^3 + 5*A*a^2*b^4)*e^4)*x^2 + (64*B*b^6*d^4 - 8*(29*B*
a*b^5 - A*b^6)*d^3*e + 4*(65*B*a^2*b^4 - 11*A*a*b^5)*d^2*e^2 - (125*B*a^3*b^3 - 109*A*a^2*b^4)*d*e^3 + (33*B*a
^4*b^2 - 73*A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d^4 - 4*a^5*b^6*d^3*e + 6*a^6*b^5*d^2*e^2 - 4*a^7*b^4*d
*e^3 + a^8*b^3*e^4 + (b^11*d^4 - 4*a*b^10*d^3*e + 6*a^2*b^9*d^2*e^2 - 4*a^3*b^8*d*e^3 + a^4*b^7*e^4)*x^4 + 4*(
a*b^10*d^4 - 4*a^2*b^9*d^3*e + 6*a^3*b^8*d^2*e^2 - 4*a^4*b^7*d*e^3 + a^5*b^6*e^4)*x^3 + 6*(a^2*b^9*d^4 - 4*a^3
*b^8*d^3*e + 6*a^4*b^7*d^2*e^2 - 4*a^5*b^6*d*e^3 + a^6*b^5*e^4)*x^2 + 4*(a^3*b^8*d^4 - 4*a^4*b^7*d^3*e + 6*a^5
*b^6*d^2*e^2 - 4*a^6*b^5*d*e^3 + a^7*b^4*e^4)*x)]

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giac [B]  time = 0.44, size = 787, normalized size = 2.19 \[ \frac {{\left (8 \, B b d e^{3} - 3 \, B a e^{4} - 5 \, A b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{5} d^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 3 \, a b^{4} d^{2} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 3 \, a^{2} b^{3} d e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{3} b^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{4} d e^{3} - 88 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{3} + 40 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{3} + 24 \, \sqrt {x e + d} B b^{4} d^{4} e^{3} - 9 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{3} e^{4} - 15 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} e^{4} + 121 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{4} + 55 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{4} - 47 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{4} - 73 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{4} - 81 \, \sqrt {x e + d} B a b^{3} d^{3} e^{4} - 15 \, \sqrt {x e + d} A b^{4} d^{3} e^{4} - 33 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{5} - 55 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{5} - 26 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{5} + 146 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{5} + 99 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{5} + 45 \, \sqrt {x e + d} A a b^{3} d^{2} e^{5} + 33 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{6} - 73 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{6} - 51 \, \sqrt {x e + d} B a^{3} b d e^{6} - 45 \, \sqrt {x e + d} A a^{2} b^{2} d e^{6} + 9 \, \sqrt {x e + d} B a^{4} e^{7} + 15 \, \sqrt {x e + d} A a^{3} b e^{7}}{192 \, {\left (b^{5} d^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 3 \, a b^{4} d^{2} e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + 3 \, a^{2} b^{3} d e^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - a^{3} b^{2} e^{3} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

1/64*(8*B*b*d*e^3 - 3*B*a*e^4 - 5*A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^3*sgn((x*e + d
)*b*e - b*d*e + a*e^2) - 3*a*b^4*d^2*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + 3*a^2*b^3*d*e^2*sgn((x*e + d)*b*e
- b*d*e + a*e^2) - a^3*b^2*e^3*sgn((x*e + d)*b*e - b*d*e + a*e^2))*sqrt(-b^2*d + a*b*e)) + 1/192*(24*(x*e + d)
^(7/2)*B*b^4*d*e^3 - 88*(x*e + d)^(5/2)*B*b^4*d^2*e^3 + 40*(x*e + d)^(3/2)*B*b^4*d^3*e^3 + 24*sqrt(x*e + d)*B*
b^4*d^4*e^3 - 9*(x*e + d)^(7/2)*B*a*b^3*e^4 - 15*(x*e + d)^(7/2)*A*b^4*e^4 + 121*(x*e + d)^(5/2)*B*a*b^3*d*e^4
+ 55*(x*e + d)^(5/2)*A*b^4*d*e^4 - 47*(x*e + d)^(3/2)*B*a*b^3*d^2*e^4 - 73*(x*e + d)^(3/2)*A*b^4*d^2*e^4 - 81
*sqrt(x*e + d)*B*a*b^3*d^3*e^4 - 15*sqrt(x*e + d)*A*b^4*d^3*e^4 - 33*(x*e + d)^(5/2)*B*a^2*b^2*e^5 - 55*(x*e +
d)^(5/2)*A*a*b^3*e^5 - 26*(x*e + d)^(3/2)*B*a^2*b^2*d*e^5 + 146*(x*e + d)^(3/2)*A*a*b^3*d*e^5 + 99*sqrt(x*e +
d)*B*a^2*b^2*d^2*e^5 + 45*sqrt(x*e + d)*A*a*b^3*d^2*e^5 + 33*(x*e + d)^(3/2)*B*a^3*b*e^6 - 73*(x*e + d)^(3/2)
*A*a^2*b^2*e^6 - 51*sqrt(x*e + d)*B*a^3*b*d*e^6 - 45*sqrt(x*e + d)*A*a^2*b^2*d*e^6 + 9*sqrt(x*e + d)*B*a^4*e^7
+ 15*sqrt(x*e + d)*A*a^3*b*e^7)/((b^5*d^3*sgn((x*e + d)*b*e - b*d*e + a*e^2) - 3*a*b^4*d^2*e*sgn((x*e + d)*b*
e - b*d*e + a*e^2) + 3*a^2*b^3*d*e^2*sgn((x*e + d)*b*e - b*d*e + a*e^2) - a^3*b^2*e^3*sgn((x*e + d)*b*e - b*d*
e + a*e^2))*((x*e + d)*b - b*d + a*e)^4)

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maple [B]  time = 0.08, size = 1296, normalized size = 3.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(b*x+a)/e*(73*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*A*a^2*b^2*e^3+9*B*a*b^4*e^5*x^4*arctan((e*x+d)^(1/2)/((a
*e-b*d)*b)^(1/2)*b)+9*B*a^5*e^5*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-24*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/
2)*B*b^4*d^4+15*A*a^4*b*e^5*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-40*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*B
*b^4*d^3+15*A*b^5*e^5*x^4*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+15*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*A*b
^4*e-24*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*B*b^4*d+88*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*B*b^4*d^2-9*((a*e-b*d)*
b)^(1/2)*(e*x+d)^(1/2)*B*a^4*e^4+45*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*A*a^2*b^2*d*e^3-45*((a*e-b*d)*b)^(1/2)*(
e*x+d)^(1/2)*A*a*b^3*d^2*e^2+51*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*B*a^3*b*d*e^3-99*((a*e-b*d)*b)^(1/2)*(e*x+d)
^(1/2)*B*a^2*b^2*d^2*e^2+81*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*B*a*b^3*d^3*e-146*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3
/2)*A*a*b^3*d*e^2+26*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*B*a^2*b^2*d*e^2+47*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*B*
a*b^3*d^2*e-96*B*a^3*b^2*d*e^4*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-144*B*a^2*b^3*d*e^4*x^2*arctan((e
*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-121*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*B*a*b^3*d*e-96*B*a*b^4*d*e^4*x^3*arct
an((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-15*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*A*a^3*b*e^4+15*((a*e-b*d)*b)^(1/2
)*(e*x+d)^(1/2)*A*b^4*d^3*e+60*A*a^3*b^2*e^5*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+33*((a*e-b*d)*b)^(1
/2)*(e*x+d)^(5/2)*B*a^2*b^2*e^2+36*B*a^4*b*e^5*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+90*A*a^2*b^3*e^5*
x^2*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+54*B*a^3*b^2*e^5*x^2*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*
b)-33*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*B*a^3*b*e^3-24*B*a^4*b*d*e^4*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*
b)+73*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*A*b^4*d^2*e+55*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*A*a*b^3*e^2-55*((a*e-
b*d)*b)^(1/2)*(e*x+d)^(5/2)*A*b^4*d*e-24*B*b^5*d*e^4*x^4*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+60*A*a*b^
4*e^5*x^3*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+36*B*a^2*b^3*e^5*x^3*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^
(1/2)*b)+9*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*B*a*b^3*e)/((a*e-b*d)*b)^(1/2)/b^2/(a*e-b*d)/(a^2*e^2-2*a*b*d*e+b
^2*d^2)/((b*x+a)^2)^(5/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,\sqrt {d+e\,x}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(1/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(1/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(1/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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