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

Optimal. Leaf size=258 \[ \frac{\sqrt{x} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a B+7 A b)}{64 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a B+7 A b)}{96 a^3 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (a B+7 A b)}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (a+b x) (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{9/2} b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(5*(7*A*b + a*B)*Sqrt[x])/(64*a^4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*Sqrt[x])/(4*a*b*(a + b*x)^3*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((7*A*b + a*B)*Sqrt[x])/(24*a^2*b*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (5*(7*A*b + a*B)*Sqrt[x])/(96*a^3*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(7*A*b + a*B)*(a + b*x)*Ar
cTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(9/2)*b^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.138365, antiderivative size = 258, normalized size of antiderivative = 1., 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} \[ \frac{\sqrt{x} (A b-a B)}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a B+7 A b)}{64 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 \sqrt{x} (a B+7 A b)}{96 a^3 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\sqrt{x} (a B+7 A b)}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (a+b x) (a B+7 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{9/2} b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(7*A*b + a*B)*Sqrt[x])/(64*a^4*b*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((A*b - a*B)*Sqrt[x])/(4*a*b*(a + b*x)^3*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((7*A*b + a*B)*Sqrt[x])/(24*a^2*b*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2])
+ (5*(7*A*b + a*B)*Sqrt[x])/(96*a^3*b*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (5*(7*A*b + a*B)*(a + b*x)*Ar
cTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(64*a^(9/2)*b^(3/2)*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}{\sqrt{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{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) \sqrt{x}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (b^2 (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \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) \sqrt{x}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(7 A b+a B) \sqrt{x}}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (5 b (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \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) \sqrt{x}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(7 A b+a B) \sqrt{x}}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (7 A b+a B) \sqrt{x}}{96 a^3 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (5 (7 A b+a B) \left (a b+b^2 x\right )\right ) \int \frac{1}{\sqrt{x} \left (a b+b^2 x\right )^2} \, dx}{64 a^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{5 (7 A b+a B) \sqrt{x}}{64 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) \sqrt{x}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(7 A b+a B) \sqrt{x}}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (7 A b+a B) \sqrt{x}}{96 a^3 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (5 (7 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^4 b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{5 (7 A b+a B) \sqrt{x}}{64 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) \sqrt{x}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(7 A b+a B) \sqrt{x}}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (7 A b+a B) \sqrt{x}}{96 a^3 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\left (5 (7 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^4 b \sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{5 (7 A b+a B) \sqrt{x}}{64 a^4 b \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(A b-a B) \sqrt{x}}{4 a b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{(7 A b+a B) \sqrt{x}}{24 a^2 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (7 A b+a B) \sqrt{x}}{96 a^3 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 (7 A b+a B) (a+b x) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{64 a^{9/2} b^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0353977, size = 77, normalized size = 0.3 \[ \frac{\sqrt{x} \left (a^4 (A b-a B)+(a+b x)^4 (a B+7 A b) \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};-\frac{b x}{a}\right )\right )}{4 a^5 b (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(a^4*(A*b - a*B) + (7*A*b + a*B)*(a + b*x)^4*Hypergeometric2F1[1/2, 4, 3/2, -((b*x)/a)]))/(4*a^5*b*(a
+ b*x)^3*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.016, size = 357, normalized size = 1.4 \begin{align*}{\frac{bx+a}{192\,{a}^{4}b} \left ( 105\,A\sqrt{ab}{x}^{7/2}{b}^{4}+15\,B\sqrt{ab}{x}^{7/2}a{b}^{3}+385\,A\sqrt{ab}{x}^{5/2}a{b}^{3}+105\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}{b}^{5}+55\,B\sqrt{ab}{x}^{5/2}{a}^{2}{b}^{2}+15\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{4}a{b}^{4}+420\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}a{b}^{4}+60\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{3}{a}^{2}{b}^{3}+511\,A\sqrt{ab}{x}^{3/2}{a}^{2}{b}^{2}+630\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{2}{b}^{3}+73\,B\sqrt{ab}{x}^{3/2}{a}^{3}b+90\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}{b}^{2}+420\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{3}{b}^{2}+60\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) x{a}^{4}b+279\,A\sqrt{ab}\sqrt{x}{a}^{3}b+105\,A\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{4}b-15\,B\sqrt{ab}\sqrt{x}{a}^{4}+15\,B\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ){a}^{5} \right ){\frac{1}{\sqrt{ab}}} \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)/(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^(1/2),x)

[Out]

1/192*(105*A*(a*b)^(1/2)*x^(7/2)*b^4+15*B*(a*b)^(1/2)*x^(7/2)*a*b^3+385*A*(a*b)^(1/2)*x^(5/2)*a*b^3+105*A*arct
an(x^(1/2)*b/(a*b)^(1/2))*x^4*b^5+55*B*(a*b)^(1/2)*x^(5/2)*a^2*b^2+15*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^4*a*b^
4+420*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a*b^4+60*B*arctan(x^(1/2)*b/(a*b)^(1/2))*x^3*a^2*b^3+511*A*(a*b)^(1/
2)*x^(3/2)*a^2*b^2+630*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x^2*a^2*b^3+73*B*(a*b)^(1/2)*x^(3/2)*a^3*b+90*B*arctan(
x^(1/2)*b/(a*b)^(1/2))*x^2*a^3*b^2+420*A*arctan(x^(1/2)*b/(a*b)^(1/2))*x*a^3*b^2+60*B*arctan(x^(1/2)*b/(a*b)^(
1/2))*x*a^4*b+279*A*(a*b)^(1/2)*x^(1/2)*a^3*b+105*A*arctan(x^(1/2)*b/(a*b)^(1/2))*a^4*b-15*B*(a*b)^(1/2)*x^(1/
2)*a^4+15*B*arctan(x^(1/2)*b/(a*b)^(1/2))*a^5)*(b*x+a)/(a*b)^(1/2)/b/a^4/((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)/(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.35113, size = 1127, normalized size = 4.37 \begin{align*} \left [-\frac{15 \,{\left (B a^{5} + 7 \, A a^{4} b +{\left (B a b^{4} + 7 \, A b^{5}\right )} x^{4} + 4 \,{\left (B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} + 6 \,{\left (B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (B a^{4} b + 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt{-a b} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right ) + 2 \,{\left (15 \, B a^{5} b - 279 \, A a^{4} b^{2} - 15 \,{\left (B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{3} - 55 \,{\left (B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{2} - 73 \,{\left (B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x\right )} \sqrt{x}}{384 \,{\left (a^{5} b^{6} x^{4} + 4 \, a^{6} b^{5} x^{3} + 6 \, a^{7} b^{4} x^{2} + 4 \, a^{8} b^{3} x + a^{9} b^{2}\right )}}, -\frac{15 \,{\left (B a^{5} + 7 \, A a^{4} b +{\left (B a b^{4} + 7 \, A b^{5}\right )} x^{4} + 4 \,{\left (B a^{2} b^{3} + 7 \, A a b^{4}\right )} x^{3} + 6 \,{\left (B a^{3} b^{2} + 7 \, A a^{2} b^{3}\right )} x^{2} + 4 \,{\left (B a^{4} b + 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right ) +{\left (15 \, B a^{5} b - 279 \, A a^{4} b^{2} - 15 \,{\left (B a^{2} b^{4} + 7 \, A a b^{5}\right )} x^{3} - 55 \,{\left (B a^{3} b^{3} + 7 \, A a^{2} b^{4}\right )} x^{2} - 73 \,{\left (B a^{4} b^{2} + 7 \, A a^{3} b^{3}\right )} x\right )} \sqrt{x}}{192 \,{\left (a^{5} b^{6} x^{4} + 4 \, a^{6} b^{5} x^{3} + 6 \, a^{7} b^{4} x^{2} + 4 \, a^{8} b^{3} x + a^{9} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/384*(15*(B*a^5 + 7*A*a^4*b + (B*a*b^4 + 7*A*b^5)*x^4 + 4*(B*a^2*b^3 + 7*A*a*b^4)*x^3 + 6*(B*a^3*b^2 + 7*A*
a^2*b^3)*x^2 + 4*(B*a^4*b + 7*A*a^3*b^2)*x)*sqrt(-a*b)*log((b*x - a - 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) + 2*(15
*B*a^5*b - 279*A*a^4*b^2 - 15*(B*a^2*b^4 + 7*A*a*b^5)*x^3 - 55*(B*a^3*b^3 + 7*A*a^2*b^4)*x^2 - 73*(B*a^4*b^2 +
7*A*a^3*b^3)*x)*sqrt(x))/(a^5*b^6*x^4 + 4*a^6*b^5*x^3 + 6*a^7*b^4*x^2 + 4*a^8*b^3*x + a^9*b^2), -1/192*(15*(B
*a^5 + 7*A*a^4*b + (B*a*b^4 + 7*A*b^5)*x^4 + 4*(B*a^2*b^3 + 7*A*a*b^4)*x^3 + 6*(B*a^3*b^2 + 7*A*a^2*b^3)*x^2 +
4*(B*a^4*b + 7*A*a^3*b^2)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (15*B*a^5*b - 279*A*a^4*b^2 - 15*(B*a^
2*b^4 + 7*A*a*b^5)*x^3 - 55*(B*a^3*b^3 + 7*A*a^2*b^4)*x^2 - 73*(B*a^4*b^2 + 7*A*a^3*b^3)*x)*sqrt(x))/(a^5*b^6*
x^4 + 4*a^6*b^5*x^3 + 6*a^7*b^4*x^2 + 4*a^8*b^3*x + a^9*b^2)]

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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)/(b**2*x**2+2*a*b*x+a**2)**(5/2)/x**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.18667, size = 198, normalized size = 0.77 \begin{align*} \frac{5 \,{\left (B a + 7 \, A b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{64 \, \sqrt{a b} a^{4} b \mathrm{sgn}\left (b x + a\right )} + \frac{15 \, B a b^{3} x^{\frac{7}{2}} + 105 \, A b^{4} x^{\frac{7}{2}} + 55 \, B a^{2} b^{2} x^{\frac{5}{2}} + 385 \, A a b^{3} x^{\frac{5}{2}} + 73 \, B a^{3} b x^{\frac{3}{2}} + 511 \, A a^{2} b^{2} x^{\frac{3}{2}} - 15 \, B a^{4} \sqrt{x} + 279 \, A a^{3} b \sqrt{x}}{192 \,{\left (b x + a\right )}^{4} a^{4} b \mathrm{sgn}\left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/64*(B*a + 7*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4*b*sgn(b*x + a)) + 1/192*(15*B*a*b^3*x^(7/2) + 10
5*A*b^4*x^(7/2) + 55*B*a^2*b^2*x^(5/2) + 385*A*a*b^3*x^(5/2) + 73*B*a^3*b*x^(3/2) + 511*A*a^2*b^2*x^(3/2) - 15
*B*a^4*sqrt(x) + 279*A*a^3*b*sqrt(x))/((b*x + a)^4*a^4*b*sgn(b*x + a))