3.17.58 \(\int \frac {(A+B x) (d+e x)^{3/2}}{(a^2+2 a b x+b^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=359 \[ -\frac {(d+e x)^{5/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{3/2} (-5 a B e-3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {e^3 (a+b x) (-5 a B e-3 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}-\frac {e^2 \sqrt {d+e x} (-5 a B e-3 A b e+8 b B d)}{64 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {e \sqrt {d+e x} (-5 a B e-3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]

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Rubi [A]  time = 0.32, 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} \begin {gather*} -\frac {e^2 \sqrt {d+e x} (-5 a B e-3 A b e+8 b B d)}{64 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}+\frac {e^3 (a+b x) (-5 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^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}-\frac {e \sqrt {d+e x} (-5 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)}-\frac {(d+e x)^{5/2} (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)^{3/2} (-5 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)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(3/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

-(e^2*(8*b*B*d - 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(64*b^3*(b*d - a*e)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (e*(
8*b*B*d - 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/(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 - 5*a*B*e)*(d + e*x)^(3/2))/(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)^(5/2))/(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 - 3*
A*b*e - 5*a*B*e)*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*b^(7/2)*(b*d - a*e)^(5/2)*Sqr
t[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) (d+e x)^{3/2}}{\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)^{3/2}}{\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)^{5/2}}{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-5 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{\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-5 a B e) (d+e x)^{3/2}}{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)^{5/2}}{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-3 A b e-5 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\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 {e (8 b B d-3 A b e-5 a B e) \sqrt {d+e x}}{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-5 a B e) (d+e x)^{3/2}}{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)^{5/2}}{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-3 A b e-5 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^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e^2 (8 b B d-3 A b e-5 a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (8 b B d-3 A b e-5 a B e) \sqrt {d+e x}}{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-5 a B e) (d+e x)^{3/2}}{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)^{5/2}}{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-3 A b e-5 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^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e^2 (8 b B d-3 A b e-5 a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (8 b B d-3 A b e-5 a B e) \sqrt {d+e x}}{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-5 a B e) (d+e x)^{3/2}}{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)^{5/2}}{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-3 A b e-5 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^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e^2 (8 b B d-3 A b e-5 a B e) \sqrt {d+e x}}{64 b^3 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (8 b B d-3 A b e-5 a B e) \sqrt {d+e x}}{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-5 a B e) (d+e x)^{3/2}}{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)^{5/2}}{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-3 A b e-5 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{7/2} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.12, size = 116, normalized size = 0.32 \begin {gather*} \frac {(d+e x)^{5/2} \left (-\frac {e^3 (a+b x)^4 (5 a B e+3 A b e-8 b B d) \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+5 a B-5 A b\right )}{20 b (a+b x)^3 \sqrt {(a+b x)^2} (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(3/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((d + e*x)^(5/2)*(-5*A*b + 5*a*B - (e^3*(-8*b*B*d + 3*A*b*e + 5*a*B*e)*(a + b*x)^4*Hypergeometric2F1[5/2, 4, 7
/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^4))/(20*b*(b*d - a*e)*(a + b*x)^3*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 58.58, size = 511, normalized size = 1.42 \begin {gather*} \frac {(-a e-b e x) \left (\frac {\left (5 a B e^4+3 A b e^4-8 b B d e^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{7/2} (b d-a e)^2 \sqrt {a e-b d}}-\frac {e^3 \sqrt {d+e x} \left (-15 a^4 B e^4-9 a^3 A b e^4-55 a^3 b B e^3 (d+e x)+69 a^3 b B d e^3-33 a^2 A b^2 e^3 (d+e x)+27 a^2 A b^2 d e^3-117 a^2 b^2 B d^2 e^2-73 a^2 b^2 B e^2 (d+e x)^2+198 a^2 b^2 B d e^2 (d+e x)-27 a A b^3 d^2 e^2+33 a A b^3 e^2 (d+e x)^2+66 a A b^3 d e^2 (d+e x)+87 a b^3 B d^3 e-231 a b^3 B d^2 e (d+e x)+15 a b^3 B e (d+e x)^3+113 a b^3 B d e (d+e x)^2+9 A b^4 d^3 e-33 A b^4 d^2 e (d+e x)+9 A b^4 e (d+e x)^3-33 A b^4 d e (d+e x)^2-24 b^4 B d^4+88 b^4 B d^3 (d+e x)-40 b^4 B d^2 (d+e x)^2-24 b^4 B d (d+e x)^3\right )}{192 b^3 (b d-a e)^2 (-a e-b (d+e x)+b d)^4}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(3/2))/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

((-(a*e) - b*e*x)*(-1/192*(e^3*Sqrt[d + e*x]*(-24*b^4*B*d^4 + 9*A*b^4*d^3*e + 87*a*b^3*B*d^3*e - 27*a*A*b^3*d^
2*e^2 - 117*a^2*b^2*B*d^2*e^2 + 27*a^2*A*b^2*d*e^3 + 69*a^3*b*B*d*e^3 - 9*a^3*A*b*e^4 - 15*a^4*B*e^4 + 88*b^4*
B*d^3*(d + e*x) - 33*A*b^4*d^2*e*(d + e*x) - 231*a*b^3*B*d^2*e*(d + e*x) + 66*a*A*b^3*d*e^2*(d + e*x) + 198*a^
2*b^2*B*d*e^2*(d + e*x) - 33*a^2*A*b^2*e^3*(d + e*x) - 55*a^3*b*B*e^3*(d + e*x) - 40*b^4*B*d^2*(d + e*x)^2 - 3
3*A*b^4*d*e*(d + e*x)^2 + 113*a*b^3*B*d*e*(d + e*x)^2 + 33*a*A*b^3*e^2*(d + e*x)^2 - 73*a^2*b^2*B*e^2*(d + e*x
)^2 - 24*b^4*B*d*(d + e*x)^3 + 9*A*b^4*e*(d + e*x)^3 + 15*a*b^3*B*e*(d + e*x)^3))/(b^3*(b*d - a*e)^2*(b*d - a*
e - b*(d + e*x))^4) + ((-8*b*B*d*e^3 + 3*A*b*e^4 + 5*a*B*e^4)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x]
)/(b*d - a*e)])/(64*b^(7/2)*(b*d - a*e)^2*Sqrt[-(b*d) + a*e])))/(e*Sqrt[(a*e + b*e*x)^2/e^2])

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fricas [B]  time = 0.46, size = 1706, normalized size = 4.75

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(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 - (5*B*a^5 + 3*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (5*B*a*b^4 + 3*A*b^5)*e^4)*x^4 + 4*
(8*B*a*b^4*d*e^3 - (5*B*a^2*b^3 + 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (5*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 - (5*B*a^4*b + 3*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 - 24*(B*a^2*b^4 + 5*A*a*b^5)
*d^3*e - 6*(B*a^3*b^3 - 13*A*a^2*b^4)*d^2*e^2 + (29*B*a^4*b^2 + 3*A*a^3*b^3)*d*e^3 - 3*(5*B*a^5*b + 3*A*a^4*b^
2)*e^4 + 3*(8*B*b^6*d^2*e^2 - (13*B*a*b^5 + 3*A*b^6)*d*e^3 + (5*B*a^2*b^4 + 3*A*a*b^5)*e^4)*x^3 + (112*B*b^6*d
^3*e - 6*(45*B*a*b^5 - A*b^6)*d^2*e^2 + 3*(77*B*a^2*b^4 - 13*A*a*b^5)*d*e^3 - (73*B*a^3*b^3 - 33*A*a^2*b^4)*e^
4)*x^2 + (64*B*b^6*d^4 - 8*(13*B*a*b^5 - 9*A*b^6)*d^3*e - 12*(B*a^2*b^4 + 17*A*a*b^5)*d^2*e^2 + (107*B*a^3*b^3
+ 165*A*a^2*b^4)*d*e^3 - 11*(5*B*a^4*b^2 + 3*A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d^3 - 3*a^5*b^6*d^2*e
+ 3*a^6*b^5*d*e^2 - a^7*b^4*e^3 + (b^11*d^3 - 3*a*b^10*d^2*e + 3*a^2*b^9*d*e^2 - a^3*b^8*e^3)*x^4 + 4*(a*b^10
*d^3 - 3*a^2*b^9*d^2*e + 3*a^3*b^8*d*e^2 - a^4*b^7*e^3)*x^3 + 6*(a^2*b^9*d^3 - 3*a^3*b^8*d^2*e + 3*a^4*b^7*d*e
^2 - a^5*b^6*e^3)*x^2 + 4*(a^3*b^8*d^3 - 3*a^4*b^7*d^2*e + 3*a^5*b^6*d*e^2 - a^6*b^5*e^3)*x), -1/192*(3*(8*B*a
^4*b*d*e^3 - (5*B*a^5 + 3*A*a^4*b)*e^4 + (8*B*b^5*d*e^3 - (5*B*a*b^4 + 3*A*b^5)*e^4)*x^4 + 4*(8*B*a*b^4*d*e^3
- (5*B*a^2*b^3 + 3*A*a*b^4)*e^4)*x^3 + 6*(8*B*a^2*b^3*d*e^3 - (5*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 - (5*B*a^4*b + 3*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 - 24*(B*a^2*b^4 + 5*A*a*b^5)*d^3*e - 6*(B*a^3*b^3 - 13*A*a^2*b^4
)*d^2*e^2 + (29*B*a^4*b^2 + 3*A*a^3*b^3)*d*e^3 - 3*(5*B*a^5*b + 3*A*a^4*b^2)*e^4 + 3*(8*B*b^6*d^2*e^2 - (13*B*
a*b^5 + 3*A*b^6)*d*e^3 + (5*B*a^2*b^4 + 3*A*a*b^5)*e^4)*x^3 + (112*B*b^6*d^3*e - 6*(45*B*a*b^5 - A*b^6)*d^2*e^
2 + 3*(77*B*a^2*b^4 - 13*A*a*b^5)*d*e^3 - (73*B*a^3*b^3 - 33*A*a^2*b^4)*e^4)*x^2 + (64*B*b^6*d^4 - 8*(13*B*a*b
^5 - 9*A*b^6)*d^3*e - 12*(B*a^2*b^4 + 17*A*a*b^5)*d^2*e^2 + (107*B*a^3*b^3 + 165*A*a^2*b^4)*d*e^3 - 11*(5*B*a^
4*b^2 + 3*A*a^3*b^3)*e^4)*x)*sqrt(e*x + d))/(a^4*b^7*d^3 - 3*a^5*b^6*d^2*e + 3*a^6*b^5*d*e^2 - a^7*b^4*e^3 + (
b^11*d^3 - 3*a*b^10*d^2*e + 3*a^2*b^9*d*e^2 - a^3*b^8*e^3)*x^4 + 4*(a*b^10*d^3 - 3*a^2*b^9*d^2*e + 3*a^3*b^8*d
*e^2 - a^4*b^7*e^3)*x^3 + 6*(a^2*b^9*d^3 - 3*a^3*b^8*d^2*e + 3*a^4*b^7*d*e^2 - a^5*b^6*e^3)*x^2 + 4*(a^3*b^8*d
^3 - 3*a^4*b^7*d^2*e + 3*a^5*b^6*d*e^2 - a^6*b^5*e^3)*x)]

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giac [B]  time = 0.45, size = 715, normalized size = 1.99 \begin {gather*} -\frac {{\left (8 \, B b d e^{3} - 5 \, B a e^{4} - 3 \, 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^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 2 \, a b^{4} d e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{2} b^{3} e^{2} \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} + 40 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{3} - 88 \, {\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} - 15 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{3} e^{4} - 9 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{4} e^{4} - 113 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{4} + 33 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{4} + 231 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{4} + 33 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{4} - 87 \, \sqrt {x e + d} B a b^{3} d^{3} e^{4} - 9 \, \sqrt {x e + d} A b^{4} d^{3} e^{4} + 73 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{5} - 33 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{5} - 198 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{5} - 66 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{5} + 117 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{5} + 27 \, \sqrt {x e + d} A a b^{3} d^{2} e^{5} + 55 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{6} + 33 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{6} - 69 \, \sqrt {x e + d} B a^{3} b d e^{6} - 27 \, \sqrt {x e + d} A a^{2} b^{2} d e^{6} + 15 \, \sqrt {x e + d} B a^{4} e^{7} + 9 \, \sqrt {x e + d} A a^{3} b e^{7}}{192 \, {\left (b^{5} d^{2} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) - 2 \, a b^{4} d e \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right ) + a^{2} b^{3} e^{2} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

-1/64*(8*B*b*d*e^3 - 5*B*a*e^4 - 3*A*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^5*d^2*sgn((x*e +
d)*b*e - b*d*e + a*e^2) - 2*a*b^4*d*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^2*b^3*e^2*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 + 40*(x*e + d)^(5/2)*B*b^4*d^2*e^3
- 88*(x*e + d)^(3/2)*B*b^4*d^3*e^3 + 24*sqrt(x*e + d)*B*b^4*d^4*e^3 - 15*(x*e + d)^(7/2)*B*a*b^3*e^4 - 9*(x*e
+ d)^(7/2)*A*b^4*e^4 - 113*(x*e + d)^(5/2)*B*a*b^3*d*e^4 + 33*(x*e + d)^(5/2)*A*b^4*d*e^4 + 231*(x*e + d)^(3/2
)*B*a*b^3*d^2*e^4 + 33*(x*e + d)^(3/2)*A*b^4*d^2*e^4 - 87*sqrt(x*e + d)*B*a*b^3*d^3*e^4 - 9*sqrt(x*e + d)*A*b^
4*d^3*e^4 + 73*(x*e + d)^(5/2)*B*a^2*b^2*e^5 - 33*(x*e + d)^(5/2)*A*a*b^3*e^5 - 198*(x*e + d)^(3/2)*B*a^2*b^2*
d*e^5 - 66*(x*e + d)^(3/2)*A*a*b^3*d*e^5 + 117*sqrt(x*e + d)*B*a^2*b^2*d^2*e^5 + 27*sqrt(x*e + d)*A*a*b^3*d^2*
e^5 + 55*(x*e + d)^(3/2)*B*a^3*b*e^6 + 33*(x*e + d)^(3/2)*A*a^2*b^2*e^6 - 69*sqrt(x*e + d)*B*a^3*b*d*e^6 - 27*
sqrt(x*e + d)*A*a^2*b^2*d*e^6 + 15*sqrt(x*e + d)*B*a^4*e^7 + 9*sqrt(x*e + d)*A*a^3*b*e^7)/((b^5*d^2*sgn((x*e +
d)*b*e - b*d*e + a*e^2) - 2*a*b^4*d*e*sgn((x*e + d)*b*e - b*d*e + a*e^2) + a^2*b^3*e^2*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 = 1273, normalized size = 3.55

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(b*x+a)/e*(-33*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*A*a^2*b^2*e^3+15*B*a*b^4*e^5*x^4*arctan((e*x+d)^(1/2)/(
(a*e-b*d)*b)^(1/2)*b)+15*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+9*A*a^4*b*e^5*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+88*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)
*B*b^4*d^3+9*A*b^5*e^5*x^4*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)*A*b
^4*e-24*((a*e-b*d)*b)^(1/2)*(e*x+d)^(7/2)*B*b^4*d-40*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*B*b^4*d^2-15*((a*e-b*d)
*b)^(1/2)*(e*x+d)^(1/2)*B*a^4*e^4+27*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*A*a^2*b^2*d*e^3-27*((a*e-b*d)*b)^(1/2)*
(e*x+d)^(1/2)*A*a*b^3*d^2*e^2+69*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*B*a^3*b*d*e^3-117*((a*e-b*d)*b)^(1/2)*(e*x+
d)^(1/2)*B*a^2*b^2*d^2*e^2+87*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*B*a*b^3*d^3*e+66*((a*e-b*d)*b)^(1/2)*(e*x+d)^(
3/2)*A*a*b^3*d*e^2+198*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*B*a^2*b^2*d*e^2-231*((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)+113*((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*a
rctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)-9*((a*e-b*d)*b)^(1/2)*(e*x+d)^(1/2)*A*a^3*b*e^4+9*((a*e-b*d)*b)^(1/
2)*(e*x+d)^(1/2)*A*b^4*d^3*e+36*A*a^3*b^2*e^5*x*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)^(5/2)*B*a^2*b^2*e^2+60*B*a^4*b*e^5*x*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+54*A*a^2*b^3*e^5
*x^2*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+90*B*a^3*b^2*e^5*x^2*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)
*b)-55*((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)-33*((a*e-b*d)*b)^(1/2)*(e*x+d)^(3/2)*A*b^4*d^2*e+33*((a*e-b*d)*b)^(1/2)*(e*x+d)^(5/2)*A*a*b^3*e^2-33*((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)+36*A*a*b
^4*e^5*x^3*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)+60*B*a^2*b^3*e^5*x^3*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)*B*a*b^3*e)/((a*e-b*d)*b)^(1/2)/b^3/(a*e-b*d)^2/((b*x+a)^2)^(5/2
)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(3/2)/(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 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(3/2))/(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(3/2))/(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 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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