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

Optimal. Leaf size=255 \[ \frac {7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {b} (a+b x) (7 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (a+b x) (7 A b-3 a B)}{4 a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (a+b x) (7 A b-3 a B)}{12 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]

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Rubi [A]  time = 0.13, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {770, 78, 51, 63, 205} \begin {gather*} \frac {5 (a+b x) (7 A b-3 a B)}{4 a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (a+b x) (7 A b-3 a B)}{12 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {b} (a+b x) (7 A b-3 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*A*b - 3*a*B)/(4*a^2*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(2*a*b*x^(3/2)*(a + b*x)*Sqrt[a^
2 + 2*a*b*x + b^2*x^2]) - (5*(7*A*b - 3*a*B)*(a + b*x))/(12*a^3*b*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*
(7*A*b - 3*a*B)*(a + b*x))/(4*a^4*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*Sqrt[b]*(7*A*b - 3*a*B)*(a + b*x
)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[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}{x^{5/2} \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}{x^{5/2} \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}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left ((7 A b-3 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \left (a b+b^2 x\right )^2} \, dx}{4 a \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 (7 A b-3 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{5/2} \left (a b+b^2 x\right )} \, dx}{8 a^2 b \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (7 A b-3 a B) (a+b x)}{12 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 (7 A b-3 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{x^{3/2} \left (a b+b^2 x\right )} \, dx}{8 a^3 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (7 A b-3 a B) (a+b x)}{12 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b-3 a B) (a+b x)}{4 a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 b (7 A b-3 a B) \left (a b+b^2 x\right )\right ) \int \frac {1}{\sqrt {x} \left (a b+b^2 x\right )} \, dx}{8 a^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (7 A b-3 a B) (a+b x)}{12 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b-3 a B) (a+b x)}{4 a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 b (7 A b-3 a B) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+b^2 x^2} \, dx,x,\sqrt {x}\right )}{4 a^4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {7 A b-3 a B}{4 a^2 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {A b-a B}{2 a b x^{3/2} (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 (7 A b-3 a B) (a+b x)}{12 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 (7 A b-3 a B) (a+b x)}{4 a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 \sqrt {b} (7 A b-3 a B) (a+b x) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 79, normalized size = 0.31 \begin {gather*} \frac {3 a^2 (A b-a B)+(a+b x)^2 (3 a B-7 A b) \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};-\frac {b x}{a}\right )}{6 a^3 b x^{3/2} (a+b x) \sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*(A*b - a*B) + (-7*A*b + 3*a*B)*(a + b*x)^2*Hypergeometric2F1[-3/2, 2, -1/2, -((b*x)/a)])/(6*a^3*b*x^(3/
2)*(a + b*x)*Sqrt[(a + b*x)^2])

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IntegrateAlgebraic [A]  time = 27.61, size = 142, normalized size = 0.56 \begin {gather*} \frac {(a+b x) \left (\frac {-8 a^3 A-24 a^3 B x+56 a^2 A b x-75 a^2 b B x^2+175 a A b^2 x^2-45 a b^2 B x^3+105 A b^3 x^3}{12 a^4 x^{3/2} (a+b x)^2}-\frac {5 \left (3 a \sqrt {b} B-7 A b^{3/2}\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}}\right )}{\sqrt {(a+b x)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x)*((-8*a^3*A + 56*a^2*A*b*x - 24*a^3*B*x + 175*a*A*b^2*x^2 - 75*a^2*b*B*x^2 + 105*A*b^3*x^3 - 45*a*b^
2*B*x^3)/(12*a^4*x^(3/2)*(a + b*x)^2) - (5*(-7*A*b^(3/2) + 3*a*Sqrt[b]*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(
4*a^(9/2))))/Sqrt[(a + b*x)^2]

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fricas [A]  time = 0.45, size = 380, normalized size = 1.49 \begin {gather*} \left [-\frac {15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {15 \, {\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + 2 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3} + {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (8 \, A a^{3} + 15 \, {\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} + 25 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 8 \, {\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/24*(15*((3*B*a*b^2 - 7*A*b^3)*x^4 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^3 + (3*B*a^3 - 7*A*a^2*b)*x^2)*sqrt(-b/a)*
log((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(8*A*a^3 + 15*(3*B*a*b^2 - 7*A*b^3)*x^3 + 25*(3*B*a^2*b
- 7*A*a*b^2)*x^2 + 8*(3*B*a^3 - 7*A*a^2*b)*x)*sqrt(x))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2), 1/12*(15*((3*B*a
*b^2 - 7*A*b^3)*x^4 + 2*(3*B*a^2*b - 7*A*a*b^2)*x^3 + (3*B*a^3 - 7*A*a^2*b)*x^2)*sqrt(b/a)*arctan(a*sqrt(b/a)/
(b*sqrt(x))) - (8*A*a^3 + 15*(3*B*a*b^2 - 7*A*b^3)*x^3 + 25*(3*B*a^2*b - 7*A*a*b^2)*x^2 + 8*(3*B*a^3 - 7*A*a^2
*b)*x)*sqrt(x))/(a^4*b^2*x^4 + 2*a^5*b*x^3 + a^6*x^2)]

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giac [A]  time = 0.21, size = 132, normalized size = 0.52 \begin {gather*} -\frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, {\left (3 \, B a x - 9 \, A b x + A a\right )}}{3 \, a^{4} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right )} - \frac {7 \, B a b^{2} x^{\frac {3}{2}} - 11 \, A b^{3} x^{\frac {3}{2}} + 9 \, B a^{2} b \sqrt {x} - 13 \, A a b^{2} \sqrt {x}}{4 \, {\left (b x + a\right )}^{2} a^{4} \mathrm {sgn}\left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/4*(3*B*a*b - 7*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*sgn(b*x + a)) - 2/3*(3*B*a*x - 9*A*b*x + A
*a)/(a^4*x^(3/2)*sgn(b*x + a)) - 1/4*(7*B*a*b^2*x^(3/2) - 11*A*b^3*x^(3/2) + 9*B*a^2*b*sqrt(x) - 13*A*a*b^2*sq
rt(x))/((b*x + a)^2*a^4*sgn(b*x + a))

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maple [A]  time = 0.07, size = 253, normalized size = 0.99 \begin {gather*} \frac {\left (105 A \,b^{4} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-45 B a \,b^{3} x^{\frac {7}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+210 A a \,b^{3} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-90 B \,a^{2} b^{2} x^{\frac {5}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+105 A \,a^{2} b^{2} x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )-45 B \,a^{3} b \,x^{\frac {3}{2}} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )+105 \sqrt {a b}\, A \,b^{3} x^{3}-45 \sqrt {a b}\, B a \,b^{2} x^{3}+175 \sqrt {a b}\, A a \,b^{2} x^{2}-75 \sqrt {a b}\, B \,a^{2} b \,x^{2}+56 \sqrt {a b}\, A \,a^{2} b x -24 \sqrt {a b}\, B \,a^{3} x -8 \sqrt {a b}\, A \,a^{3}\right ) \left (b x +a \right )}{12 \sqrt {a b}\, \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} a^{4} x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(105*(a*b)^(1/2)*A*b^3*x^3+105*A*b^4*x^(7/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))-45*(a*b)^(1/2)*B*a*b^2*x^3-4
5*B*a*b^3*x^(7/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))+210*A*x^(5/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*a*b^3-90*B*x^(
5/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*a^2*b^2+175*(a*b)^(1/2)*A*a*b^2*x^2+105*A*x^(3/2)*arctan(1/(a*b)^(1/2)*b*
x^(1/2))*a^2*b^2-75*(a*b)^(1/2)*B*a^2*b*x^2-45*B*x^(3/2)*arctan(1/(a*b)^(1/2)*b*x^(1/2))*a^3*b+56*(a*b)^(1/2)*
A*a^2*b*x-24*(a*b)^(1/2)*B*a^3*x-8*(a*b)^(1/2)*A*a^3)*(b*x+a)/x^(3/2)/(a*b)^(1/2)/a^4/((b*x+a)^2)^(3/2)

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maxima [A]  time = 1.66, size = 341, normalized size = 1.34 \begin {gather*} -\frac {420 \, {\left (B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{\frac {5}{2}} + 5 \, {\left ({\left (B a b^{5} - 7 \, A b^{6}\right )} x^{2} + 21 \, {\left (B a^{2} b^{4} - 3 \, A a b^{5}\right )} x\right )} x^{\frac {5}{2}} - 5 \, {\left (9 \, {\left (B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{2} - 119 \, {\left (B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x\right )} \sqrt {x} - \frac {16 \, {\left (5 \, {\left (B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{2} - 21 \, {\left (B a^{5} b - 3 \, A a^{4} b^{2}\right )} x\right )}}{\sqrt {x}} - \frac {48 \, {\left ({\left (B a^{5} b - 7 \, A a^{4} b^{2}\right )} x^{2} - {\left (B a^{6} - 3 \, A a^{5} b\right )} x\right )}}{x^{\frac {3}{2}}} + \frac {16 \, {\left (3 \, A a^{5} b x^{2} + A a^{6} x\right )}}{x^{\frac {5}{2}}}}{24 \, {\left (a^{6} b^{3} x^{3} + 3 \, a^{7} b^{2} x^{2} + 3 \, a^{8} b x + a^{9}\right )}} - \frac {5 \, {\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{4}} + \frac {5 \, {\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} \sqrt {x}\right )}}{24 \, a^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(420*(B*a^3*b^3 - 3*A*a^2*b^4)*x^(5/2) + 5*((B*a*b^5 - 7*A*b^6)*x^2 + 21*(B*a^2*b^4 - 3*A*a*b^5)*x)*x^(5
/2) - 5*(9*(B*a^3*b^3 - 7*A*a^2*b^4)*x^2 - 119*(B*a^4*b^2 - 3*A*a^3*b^3)*x)*sqrt(x) - 16*(5*(B*a^4*b^2 - 7*A*a
^3*b^3)*x^2 - 21*(B*a^5*b - 3*A*a^4*b^2)*x)/sqrt(x) - 48*((B*a^5*b - 7*A*a^4*b^2)*x^2 - (B*a^6 - 3*A*a^5*b)*x)
/x^(3/2) + 16*(3*A*a^5*b*x^2 + A*a^6*x)/x^(5/2))/(a^6*b^3*x^3 + 3*a^7*b^2*x^2 + 3*a^8*b*x + a^9) - 5/4*(3*B*a*
b - 7*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + 5/24*((B*a*b^2 - 7*A*b^3)*x^(3/2) + 6*(3*B*a^2*b -
7*A*a*b^2)*sqrt(x))/a^6

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{x^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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