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

Optimal. Leaf size=250 \[ -\frac {5 e^2 (-7 a B e+A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2} \sqrt {b d-a e}}+\frac {5 e^2 \sqrt {d+e x} (-7 a B e+A b e+6 b B d)}{8 b^4 (b d-a e)}-\frac {5 e (d+e x)^{3/2} (-7 a B e+A b e+6 b B d)}{24 b^3 (a+b x) (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e+A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 50, 63, 208} \begin {gather*} \frac {5 e^2 \sqrt {d+e x} (-7 a B e+A b e+6 b B d)}{8 b^4 (b d-a e)}-\frac {5 e^2 (-7 a B e+A b e+6 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2} \sqrt {b d-a e}}-\frac {(d+e x)^{5/2} (-7 a B e+A b e+6 b B d)}{12 b^2 (a+b x)^2 (b d-a e)}-\frac {5 e (d+e x)^{3/2} (-7 a B e+A b e+6 b B d)}{24 b^3 (a+b x) (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*e^2*(6*b*B*d + A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(8*b^4*(b*d - a*e)) - (5*e*(6*b*B*d + A*b*e - 7*a*B*e)*(d +
e*x)^(3/2))/(24*b^3*(b*d - a*e)*(a + b*x)) - ((6*b*B*d + A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(12*b^2*(b*d - a*e)
*(a + b*x)^2) - ((A*b - a*B)*(d + e*x)^(7/2))/(3*b*(b*d - a*e)*(a + b*x)^3) - (5*e^2*(6*b*B*d + A*b*e - 7*a*B*
e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*b^(9/2)*Sqrt[b*d - a*e])

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 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]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^4} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(6 b B d+A b e-7 a B e) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{6 b (b d-a e)}\\ &=-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(5 e (6 b B d+A b e-7 a B e)) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{24 b^2 (b d-a e)}\\ &=-\frac {5 e (6 b B d+A b e-7 a B e) (d+e x)^{3/2}}{24 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (5 e^2 (6 b B d+A b e-7 a B e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^3 (b d-a e)}\\ &=\frac {5 e^2 (6 b B d+A b e-7 a B e) \sqrt {d+e x}}{8 b^4 (b d-a e)}-\frac {5 e (6 b B d+A b e-7 a B e) (d+e x)^{3/2}}{24 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (5 e^2 (6 b B d+A b e-7 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^4}\\ &=\frac {5 e^2 (6 b B d+A b e-7 a B e) \sqrt {d+e x}}{8 b^4 (b d-a e)}-\frac {5 e (6 b B d+A b e-7 a B e) (d+e x)^{3/2}}{24 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(5 e (6 b B d+A b e-7 a B e)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b^4}\\ &=\frac {5 e^2 (6 b B d+A b e-7 a B e) \sqrt {d+e x}}{8 b^4 (b d-a e)}-\frac {5 e (6 b B d+A b e-7 a B e) (d+e x)^{3/2}}{24 b^3 (b d-a e) (a+b x)}-\frac {(6 b B d+A b e-7 a B e) (d+e x)^{5/2}}{12 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{7/2}}{3 b (b d-a e) (a+b x)^3}-\frac {5 e^2 (6 b B d+A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2} \sqrt {b d-a e}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.08, size = 99, normalized size = 0.40 \begin {gather*} \frac {(d+e x)^{7/2} \left (\frac {7 a B-7 A b}{(a+b x)^3}-\frac {e^2 (-7 a B e+A b e+6 b B d) \, _2F_1\left (3,\frac {7}{2};\frac {9}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}\right )}{21 b (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(7/2)*((-7*A*b + 7*a*B)/(a + b*x)^3 - (e^2*(6*b*B*d + A*b*e - 7*a*B*e)*Hypergeometric2F1[3, 7/2, 9/
2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^3))/(21*b*(b*d - a*e))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.76, size = 317, normalized size = 1.27 \begin {gather*} -\frac {e^2 \sqrt {d+e x} \left (-105 a^3 B e^3+15 a^2 A b e^3-280 a^2 b B e^2 (d+e x)+300 a^2 b B d e^2+40 a A b^2 e^2 (d+e x)-30 a A b^2 d e^2-285 a b^2 B d^2 e-231 a b^2 B e (d+e x)^2+520 a b^2 B d e (d+e x)+15 A b^3 d^2 e+33 A b^3 e (d+e x)^2-40 A b^3 d e (d+e x)+90 b^3 B d^3-240 b^3 B d^2 (d+e x)-48 b^3 B (d+e x)^3+198 b^3 B d (d+e x)^2\right )}{24 b^4 (a e+b (d+e x)-b d)^3}-\frac {5 \left (-7 a B e^3+A b e^3+6 b B d e^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{8 b^{9/2} \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/24*(e^2*Sqrt[d + e*x]*(90*b^3*B*d^3 + 15*A*b^3*d^2*e - 285*a*b^2*B*d^2*e - 30*a*A*b^2*d*e^2 + 300*a^2*b*B*d
*e^2 + 15*a^2*A*b*e^3 - 105*a^3*B*e^3 - 240*b^3*B*d^2*(d + e*x) - 40*A*b^3*d*e*(d + e*x) + 520*a*b^2*B*d*e*(d
+ e*x) + 40*a*A*b^2*e^2*(d + e*x) - 280*a^2*b*B*e^2*(d + e*x) + 198*b^3*B*d*(d + e*x)^2 + 33*A*b^3*e*(d + e*x)
^2 - 231*a*b^2*B*e*(d + e*x)^2 - 48*b^3*B*(d + e*x)^3))/(b^4*(-(b*d) + a*e + b*(d + e*x))^3) - (5*(6*b*B*d*e^2
+ A*b*e^3 - 7*a*B*e^3)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]*Sqrt[d + e*x])/(b*d - a*e)])/(8*b^(9/2)*Sqrt[-(b*d)
+ a*e])

________________________________________________________________________________________

fricas [B]  time = 0.46, size = 1075, normalized size = 4.30 \begin {gather*} \left [\frac {15 \, {\left (6 \, B a^{3} b d e^{2} - {\left (7 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (6 \, B b^{4} d e^{2} - {\left (7 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, B a b^{3} d e^{2} - {\left (7 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (6 \, B a^{2} b^{2} d e^{2} - {\left (7 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (4 \, {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{3} + 2 \, {\left (8 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e - 5 \, {\left (25 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{2} + 15 \, {\left (7 \, B a^{4} b - A a^{3} b^{2}\right )} e^{3} - 48 \, {\left (B b^{5} d e^{2} - B a b^{4} e^{3}\right )} x^{3} + 3 \, {\left (18 \, B b^{5} d^{2} e - {\left (95 \, B a b^{4} - 11 \, A b^{5}\right )} d e^{2} + 11 \, {\left (7 \, B a^{2} b^{3} - A a b^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (6 \, B b^{5} d^{3} + {\left (23 \, B a b^{4} + 13 \, A b^{5}\right )} d^{2} e - {\left (169 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} d e^{2} + 20 \, {\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (a^{3} b^{6} d - a^{4} b^{5} e + {\left (b^{9} d - a b^{8} e\right )} x^{3} + 3 \, {\left (a b^{8} d - a^{2} b^{7} e\right )} x^{2} + 3 \, {\left (a^{2} b^{7} d - a^{3} b^{6} e\right )} x\right )}}, \frac {15 \, {\left (6 \, B a^{3} b d e^{2} - {\left (7 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (6 \, B b^{4} d e^{2} - {\left (7 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (6 \, B a b^{3} d e^{2} - {\left (7 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (6 \, B a^{2} b^{2} d e^{2} - {\left (7 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (4 \, {\left (B a b^{4} + 2 \, A b^{5}\right )} d^{3} + 2 \, {\left (8 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e - 5 \, {\left (25 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{2} + 15 \, {\left (7 \, B a^{4} b - A a^{3} b^{2}\right )} e^{3} - 48 \, {\left (B b^{5} d e^{2} - B a b^{4} e^{3}\right )} x^{3} + 3 \, {\left (18 \, B b^{5} d^{2} e - {\left (95 \, B a b^{4} - 11 \, A b^{5}\right )} d e^{2} + 11 \, {\left (7 \, B a^{2} b^{3} - A a b^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (6 \, B b^{5} d^{3} + {\left (23 \, B a b^{4} + 13 \, A b^{5}\right )} d^{2} e - {\left (169 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} d e^{2} + 20 \, {\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (a^{3} b^{6} d - a^{4} b^{5} e + {\left (b^{9} d - a b^{8} e\right )} x^{3} + 3 \, {\left (a b^{8} d - a^{2} b^{7} e\right )} x^{2} + 3 \, {\left (a^{2} b^{7} d - a^{3} b^{6} e\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*(6*B*a^3*b*d*e^2 - (7*B*a^4 - A*a^3*b)*e^3 + (6*B*b^4*d*e^2 - (7*B*a*b^3 - A*b^4)*e^3)*x^3 + 3*(6*B*
a*b^3*d*e^2 - (7*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 3*(6*B*a^2*b^2*d*e^2 - (7*B*a^3*b - A*a^2*b^2)*e^3)*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*(4*(B*a*b^4 + 2*A
*b^5)*d^3 + 2*(8*B*a^2*b^3 + A*a*b^4)*d^2*e - 5*(25*B*a^3*b^2 - A*a^2*b^3)*d*e^2 + 15*(7*B*a^4*b - A*a^3*b^2)*
e^3 - 48*(B*b^5*d*e^2 - B*a*b^4*e^3)*x^3 + 3*(18*B*b^5*d^2*e - (95*B*a*b^4 - 11*A*b^5)*d*e^2 + 11*(7*B*a^2*b^3
- A*a*b^4)*e^3)*x^2 + 2*(6*B*b^5*d^3 + (23*B*a*b^4 + 13*A*b^5)*d^2*e - (169*B*a^2*b^3 - 7*A*a*b^4)*d*e^2 + 20
*(7*B*a^3*b^2 - A*a^2*b^3)*e^3)*x)*sqrt(e*x + d))/(a^3*b^6*d - a^4*b^5*e + (b^9*d - a*b^8*e)*x^3 + 3*(a*b^8*d
- a^2*b^7*e)*x^2 + 3*(a^2*b^7*d - a^3*b^6*e)*x), 1/24*(15*(6*B*a^3*b*d*e^2 - (7*B*a^4 - A*a^3*b)*e^3 + (6*B*b^
4*d*e^2 - (7*B*a*b^3 - A*b^4)*e^3)*x^3 + 3*(6*B*a*b^3*d*e^2 - (7*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 3*(6*B*a^2*b^
2*d*e^2 - (7*B*a^3*b - A*a^2*b^2)*e^3)*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)) - (4*(B*a*b^4 + 2*A*b^5)*d^3 + 2*(8*B*a^2*b^3 + A*a*b^4)*d^2*e - 5*(25*B*a^3*b^2 - A*a^2*b^3)*d*e^2
+ 15*(7*B*a^4*b - A*a^3*b^2)*e^3 - 48*(B*b^5*d*e^2 - B*a*b^4*e^3)*x^3 + 3*(18*B*b^5*d^2*e - (95*B*a*b^4 - 11*A
*b^5)*d*e^2 + 11*(7*B*a^2*b^3 - A*a*b^4)*e^3)*x^2 + 2*(6*B*b^5*d^3 + (23*B*a*b^4 + 13*A*b^5)*d^2*e - (169*B*a^
2*b^3 - 7*A*a*b^4)*d*e^2 + 20*(7*B*a^3*b^2 - A*a^2*b^3)*e^3)*x)*sqrt(e*x + d))/(a^3*b^6*d - a^4*b^5*e + (b^9*d
- a*b^8*e)*x^3 + 3*(a*b^8*d - a^2*b^7*e)*x^2 + 3*(a^2*b^7*d - a^3*b^6*e)*x)]

________________________________________________________________________________________

giac [A]  time = 0.27, size = 370, normalized size = 1.48 \begin {gather*} \frac {2 \, \sqrt {x e + d} B e^{2}}{b^{4}} + \frac {5 \, {\left (6 \, B b d e^{2} - 7 \, B a e^{3} + A b e^{3}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{4}} - \frac {54 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{3} d e^{2} - 96 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{3} d^{2} e^{2} + 42 \, \sqrt {x e + d} B b^{3} d^{3} e^{2} - 87 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{2} e^{3} + 33 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{3} e^{3} + 232 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{2} d e^{3} - 40 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{3} d e^{3} - 141 \, \sqrt {x e + d} B a b^{2} d^{2} e^{3} + 15 \, \sqrt {x e + d} A b^{3} d^{2} e^{3} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b e^{4} + 40 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{2} e^{4} + 156 \, \sqrt {x e + d} B a^{2} b d e^{4} - 30 \, \sqrt {x e + d} A a b^{2} d e^{4} - 57 \, \sqrt {x e + d} B a^{3} e^{5} + 15 \, \sqrt {x e + d} A a^{2} b e^{5}}{24 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x*e + d)*B*e^2/b^4 + 5/8*(6*B*b*d*e^2 - 7*B*a*e^3 + A*b*e^3)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e
))/(sqrt(-b^2*d + a*b*e)*b^4) - 1/24*(54*(x*e + d)^(5/2)*B*b^3*d*e^2 - 96*(x*e + d)^(3/2)*B*b^3*d^2*e^2 + 42*s
qrt(x*e + d)*B*b^3*d^3*e^2 - 87*(x*e + d)^(5/2)*B*a*b^2*e^3 + 33*(x*e + d)^(5/2)*A*b^3*e^3 + 232*(x*e + d)^(3/
2)*B*a*b^2*d*e^3 - 40*(x*e + d)^(3/2)*A*b^3*d*e^3 - 141*sqrt(x*e + d)*B*a*b^2*d^2*e^3 + 15*sqrt(x*e + d)*A*b^3
*d^2*e^3 - 136*(x*e + d)^(3/2)*B*a^2*b*e^4 + 40*(x*e + d)^(3/2)*A*a*b^2*e^4 + 156*sqrt(x*e + d)*B*a^2*b*d*e^4
- 30*sqrt(x*e + d)*A*a*b^2*d*e^4 - 57*sqrt(x*e + d)*B*a^3*e^5 + 15*sqrt(x*e + d)*A*a^2*b*e^5)/(((x*e + d)*b -
b*d + a*e)^3*b^4)

________________________________________________________________________________________

maple [B]  time = 0.08, size = 573, normalized size = 2.29 \begin {gather*} -\frac {5 \sqrt {e x +d}\, A \,a^{2} e^{5}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {5 \sqrt {e x +d}\, A a d \,e^{4}}{4 \left (b e x +a e \right )^{3} b^{2}}-\frac {5 \sqrt {e x +d}\, A \,d^{2} e^{3}}{8 \left (b e x +a e \right )^{3} b}+\frac {19 \sqrt {e x +d}\, B \,a^{3} e^{5}}{8 \left (b e x +a e \right )^{3} b^{4}}-\frac {13 \sqrt {e x +d}\, B \,a^{2} d \,e^{4}}{2 \left (b e x +a e \right )^{3} b^{3}}+\frac {47 \sqrt {e x +d}\, B a \,d^{2} e^{3}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {7 \sqrt {e x +d}\, B \,d^{3} e^{2}}{4 \left (b e x +a e \right )^{3} b}-\frac {5 \left (e x +d \right )^{\frac {3}{2}} A a \,e^{4}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {5 \left (e x +d \right )^{\frac {3}{2}} A d \,e^{3}}{3 \left (b e x +a e \right )^{3} b}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} e^{4}}{3 \left (b e x +a e \right )^{3} b^{3}}-\frac {29 \left (e x +d \right )^{\frac {3}{2}} B a d \,e^{3}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {4 \left (e x +d \right )^{\frac {3}{2}} B \,d^{2} e^{2}}{\left (b e x +a e \right )^{3} b}-\frac {11 \left (e x +d \right )^{\frac {5}{2}} A \,e^{3}}{8 \left (b e x +a e \right )^{3} b}+\frac {29 \left (e x +d \right )^{\frac {5}{2}} B a \,e^{3}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {9 \left (e x +d \right )^{\frac {5}{2}} B d \,e^{2}}{4 \left (b e x +a e \right )^{3} b}+\frac {5 A \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{3}}-\frac {35 B a \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {15 B d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {2 \sqrt {e x +d}\, B \,e^{2}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^2*B/b^4*(e*x+d)^(1/2)-11/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A+29/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*
a-9/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*d-5/3*e^4/b^2/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a+5/3*e^3/b/(b*e*x+a*e)^
3*A*(e*x+d)^(3/2)*d+17/3*e^4/b^3/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^2-29/3*e^3/b^2/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*
a*d+4*e^2/b/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^2-5/8*e^5/b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^2+5/4*e^4/b^2/(b*e*x
+a*e)^3*(e*x+d)^(1/2)*A*a*d-5/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^2+19/8*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/
2)*B*a^3-13/2*e^4/b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*d+47/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*d^2-7/4
*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^3+5/8*e^3/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1
/2)*b)*A-35/8*e^3/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*B+15/4*e^2/b^3/((a*e-b
*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d positive or negative?

________________________________________________________________________________________

mupad [B]  time = 0.23, size = 416, normalized size = 1.66 \begin {gather*} \frac {2\,B\,e^2\,\sqrt {d+e\,x}}{b^4}-\frac {{\left (d+e\,x\right )}^{5/2}\,\left (\frac {11\,A\,b^3\,e^3}{8}+\frac {9\,B\,d\,b^3\,e^2}{4}-\frac {29\,B\,a\,b^2\,e^3}{8}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (\frac {17\,B\,a^2\,b\,e^4}{3}-\frac {29\,B\,a\,b^2\,d\,e^3}{3}-\frac {5\,A\,a\,b^2\,e^4}{3}+4\,B\,b^3\,d^2\,e^2+\frac {5\,A\,b^3\,d\,e^3}{3}\right )+\sqrt {d+e\,x}\,\left (-\frac {19\,B\,a^3\,e^5}{8}+\frac {13\,B\,a^2\,b\,d\,e^4}{2}+\frac {5\,A\,a^2\,b\,e^5}{8}-\frac {47\,B\,a\,b^2\,d^2\,e^3}{8}-\frac {5\,A\,a\,b^2\,d\,e^4}{4}+\frac {7\,B\,b^3\,d^3\,e^2}{4}+\frac {5\,A\,b^3\,d^2\,e^3}{8}\right )}{b^7\,{\left (d+e\,x\right )}^3-\left (3\,b^7\,d-3\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^5\,e^2-6\,a\,b^6\,d\,e+3\,b^7\,d^2\right )-b^7\,d^3+a^3\,b^4\,e^3-3\,a^2\,b^5\,d\,e^2+3\,a\,b^6\,d^2\,e}+\frac {5\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^2\,\sqrt {d+e\,x}\,\left (A\,b\,e-7\,B\,a\,e+6\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (A\,b\,e^3-7\,B\,a\,e^3+6\,B\,b\,d\,e^2\right )}\right )\,\left (A\,b\,e-7\,B\,a\,e+6\,B\,b\,d\right )}{8\,b^{9/2}\,\sqrt {a\,e-b\,d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*e^2*(d + e*x)^(1/2))/b^4 - ((d + e*x)^(5/2)*((11*A*b^3*e^3)/8 - (29*B*a*b^2*e^3)/8 + (9*B*b^3*d*e^2)/4) -
(d + e*x)^(3/2)*((17*B*a^2*b*e^4)/3 - (5*A*a*b^2*e^4)/3 + (5*A*b^3*d*e^3)/3 + 4*B*b^3*d^2*e^2 - (29*B*a*b^2*d
*e^3)/3) + (d + e*x)^(1/2)*((5*A*a^2*b*e^5)/8 - (19*B*a^3*e^5)/8 + (5*A*b^3*d^2*e^3)/8 + (7*B*b^3*d^3*e^2)/4 -
(47*B*a*b^2*d^2*e^3)/8 - (5*A*a*b^2*d*e^4)/4 + (13*B*a^2*b*d*e^4)/2))/(b^7*(d + e*x)^3 - (3*b^7*d - 3*a*b^6*e
)*(d + e*x)^2 + (d + e*x)*(3*b^7*d^2 + 3*a^2*b^5*e^2 - 6*a*b^6*d*e) - b^7*d^3 + a^3*b^4*e^3 - 3*a^2*b^5*d*e^2
+ 3*a*b^6*d^2*e) + (5*e^2*atan((b^(1/2)*e^2*(d + e*x)^(1/2)*(A*b*e - 7*B*a*e + 6*B*b*d))/((a*e - b*d)^(1/2)*(A
*b*e^3 - 7*B*a*e^3 + 6*B*b*d*e^2)))*(A*b*e - 7*B*a*e + 6*B*b*d))/(8*b^(9/2)*(a*e - b*d)^(1/2))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________