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

Optimal. Leaf size=311 \[ \frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (9 A b-a B)}{64 a^5 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (9 A b-a B)}{192 a^4 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{11/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(35*(9*A*b - a*B))/(192*a^4*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(4*a*b*Sqrt[x]*(a + b*x)^3*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (9*A*b - a*B)/(24*a^2*b*Sqrt[x]*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(7*(9*A*b - a*B))/(96*a^3*b*Sqrt[x]*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a + b*x))/(6
4*a^5*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]
)/(64*a^(11/2)*Sqrt[b]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.158396, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (9 A b-a B)}{64 a^5 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{35 (9 A b-a B)}{192 a^4 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (a+b x) (9 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{11/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*(9*A*b - a*B))/(192*a^4*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (A*b - a*B)/(4*a*b*Sqrt[x]*(a + b*x)^3*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (9*A*b - a*B)/(24*a^2*b*Sqrt[x]*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) +
(7*(9*A*b - a*B))/(96*a^3*b*Sqrt[x]*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a + b*x))/(6
4*a^5*b*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (35*(9*A*b - a*B)*(a + b*x)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]]
)/(64*a^(11/2)*Sqrt[b]*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 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 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]

Rubi steps

\begin{align*} \int \frac{A+B x}{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}{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}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{x^{3/2} \left (a b+b^2 x\right )^4} \, dx}{8 a \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (7 b (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{x^{3/2} \left (a b+b^2 x\right )^3} \, dx}{48 a^2 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{x^{3/2} \left (a b+b^2 x\right )^2} \, dx}{192 a^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 (9 A b-a B)}{192 a^4 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{x^{3/2} \left (a b+b^2 x\right )} \, dx}{128 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 (9 A b-a B)}{192 a^4 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 (9 A b-a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )} \, dx}{128 a^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 (9 A b-a B)}{192 a^4 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{\left (35 (9 A b-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 )}{64 a^5 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{35 (9 A b-a B)}{192 a^4 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{A b-a B}{4 a b \sqrt{x} (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{9 A b-a B}{24 a^2 b \sqrt{x} (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{7 (9 A b-a B)}{96 a^3 b \sqrt{x} (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (9 A b-a B) (a+b x)}{64 a^5 b \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{35 (9 A b-a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{11/2} \sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0354171, size = 79, normalized size = 0.25 \[ \frac{a^4 (A b-a B)-(a+b x)^4 (9 A b-a B) \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};-\frac{b x}{a}\right )}{4 a^5 b \sqrt{x} (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(A*b - a*B) - (9*A*b - a*B)*(a + b*x)^4*Hypergeometric2F1[-1/2, 4, 1/2, -((b*x)/a)])/(4*a^5*b*Sqrt[x]*(a
+ b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.023, size = 374, normalized size = 1.2 \begin{align*} -{\frac{bx+a}{192\,{a}^{5}} \left ( 945\,A\sqrt{ab}{x}^{4}{b}^{4}-105\,B\sqrt{ab}{x}^{4}a{b}^{3}+3465\,A\sqrt{ab}{x}^{3}a{b}^{3}+945\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{9/2}{b}^{5}-385\,B\sqrt{ab}{x}^{3}{a}^{2}{b}^{2}-105\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{9/2}a{b}^{4}+3780\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{7/2}a{b}^{4}-420\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{7/2}{a}^{2}{b}^{3}+4599\,A\sqrt{ab}{x}^{2}{a}^{2}{b}^{2}+5670\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{2}{b}^{3}-511\,B\sqrt{ab}{x}^{2}{a}^{3}b-630\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{5/2}{a}^{3}{b}^{2}+3780\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{3}{b}^{2}-420\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3/2}{a}^{4}b+2511\,A\sqrt{ab}x{a}^{3}b+945\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) \sqrt{x}{a}^{4}b-279\,B\sqrt{ab}x{a}^{4}-105\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) \sqrt{x}{a}^{5}+384\,A\sqrt{ab}{a}^{4} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{x}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(945*A*(a*b)^(1/2)*x^4*b^4-105*B*(a*b)^(1/2)*x^4*a*b^3+3465*A*(a*b)^(1/2)*x^3*a*b^3+945*A*arctan(x^(1/2
)*b/(a*b)^(1/2))*x^(9/2)*b^5-385*B*(a*b)^(1/2)*x^3*a^2*b^2-105*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(9/2)*a*b^4+3
780*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(7/2)*a*b^4-420*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(7/2)*a^2*b^3+4599*A*(
a*b)^(1/2)*x^2*a^2*b^2+5670*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*a^2*b^3-511*B*(a*b)^(1/2)*x^2*a^3*b-630*B*
arctan(x^(1/2)*b/(a*b)^(1/2))*x^(5/2)*a^3*b^2+3780*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(3/2)*a^3*b^2-420*B*arcta
n(x^(1/2)*b/(a*b)^(1/2))*x^(3/2)*a^4*b+2511*A*(a*b)^(1/2)*x*a^3*b+945*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(1/2)*
a^4*b-279*B*(a*b)^(1/2)*x*a^4-105*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^(1/2)*a^5+384*A*(a*b)^(1/2)*a^4)*(b*x+a)/(
a*b)^(1/2)/x^(1/2)/a^5/((b*x+a)^2)^(5/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36416, size = 1212, normalized size = 3.9 \begin{align*} \left [\frac{105 \,{\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \,{\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \,{\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \,{\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a + 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) - 2 \,{\left (384 \, A a^{5} b - 105 \,{\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \,{\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \,{\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \,{\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt{x}}{384 \,{\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}, -\frac{105 \,{\left ({\left (B a b^{4} - 9 \, A b^{5}\right )} x^{5} + 4 \,{\left (B a^{2} b^{3} - 9 \, A a b^{4}\right )} x^{4} + 6 \,{\left (B a^{3} b^{2} - 9 \, A a^{2} b^{3}\right )} x^{3} + 4 \,{\left (B a^{4} b - 9 \, A a^{3} b^{2}\right )} x^{2} +{\left (B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (384 \, A a^{5} b - 105 \,{\left (B a^{2} b^{4} - 9 \, A a b^{5}\right )} x^{4} - 385 \,{\left (B a^{3} b^{3} - 9 \, A a^{2} b^{4}\right )} x^{3} - 511 \,{\left (B a^{4} b^{2} - 9 \, A a^{3} b^{3}\right )} x^{2} - 279 \,{\left (B a^{5} b - 9 \, A a^{4} b^{2}\right )} x\right )} \sqrt{x}}{192 \,{\left (a^{6} b^{5} x^{5} + 4 \, a^{7} b^{4} x^{4} + 6 \, a^{8} b^{3} x^{3} + 4 \, a^{9} b^{2} x^{2} + a^{10} b x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/384*(105*((B*a*b^4 - 9*A*b^5)*x^5 + 4*(B*a^2*b^3 - 9*A*a*b^4)*x^4 + 6*(B*a^3*b^2 - 9*A*a^2*b^3)*x^3 + 4*(B*
a^4*b - 9*A*a^3*b^2)*x^2 + (B*a^5 - 9*A*a^4*b)*x)*sqrt(-a*b)*log((b*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) -
2*(384*A*a^5*b - 105*(B*a^2*b^4 - 9*A*a*b^5)*x^4 - 385*(B*a^3*b^3 - 9*A*a^2*b^4)*x^3 - 511*(B*a^4*b^2 - 9*A*a
^3*b^3)*x^2 - 279*(B*a^5*b - 9*A*a^4*b^2)*x)*sqrt(x))/(a^6*b^5*x^5 + 4*a^7*b^4*x^4 + 6*a^8*b^3*x^3 + 4*a^9*b^2
*x^2 + a^10*b*x), -1/192*(105*((B*a*b^4 - 9*A*b^5)*x^5 + 4*(B*a^2*b^3 - 9*A*a*b^4)*x^4 + 6*(B*a^3*b^2 - 9*A*a^
2*b^3)*x^3 + 4*(B*a^4*b - 9*A*a^3*b^2)*x^2 + (B*a^5 - 9*A*a^4*b)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) +
(384*A*a^5*b - 105*(B*a^2*b^4 - 9*A*a*b^5)*x^4 - 385*(B*a^3*b^3 - 9*A*a^2*b^4)*x^3 - 511*(B*a^4*b^2 - 9*A*a^3*
b^3)*x^2 - 279*(B*a^5*b - 9*A*a^4*b^2)*x)*sqrt(x))/(a^6*b^5*x^5 + 4*a^7*b^4*x^4 + 6*a^8*b^3*x^3 + 4*a^9*b^2*x^
2 + a^10*b*x)]

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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)/x**(3/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.16138, size = 213, normalized size = 0.68 \begin{align*} \frac{35 \,{\left (B a - 9 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a^{5} \mathrm{sgn}\left (b x + a\right )} - \frac{2 \, A}{a^{5} \sqrt{x} \mathrm{sgn}\left (b x + a\right )} + \frac{105 \, B a b^{3} x^{\frac{7}{2}} - 561 \, A b^{4} x^{\frac{7}{2}} + 385 \, B a^{2} b^{2} x^{\frac{5}{2}} - 1929 \, A a b^{3} x^{\frac{5}{2}} + 511 \, B a^{3} b x^{\frac{3}{2}} - 2295 \, A a^{2} b^{2} x^{\frac{3}{2}} + 279 \, B a^{4} \sqrt{x} - 975 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a^{5} \mathrm{sgn}\left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

35/64*(B*a - 9*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5*sgn(b*x + a)) - 2*A/(a^5*sqrt(x)*sgn(b*x + a))
+ 1/192*(105*B*a*b^3*x^(7/2) - 561*A*b^4*x^(7/2) + 385*B*a^2*b^2*x^(5/2) - 1929*A*a*b^3*x^(5/2) + 511*B*a^3*b*
x^(3/2) - 2295*A*a^2*b^2*x^(3/2) + 279*B*a^4*sqrt(x) - 975*A*a^3*b*sqrt(x))/((b*x + a)^4*a^5*sgn(b*x + a))