3.765 \(\int \frac{A+B x}{x^{9/2} (a^2+2 a b x+b^2 x^2)} \, dx\)

Optimal. Leaf size=153 \[ \frac{b^2 (9 A b-7 a B)}{a^5 \sqrt{x}}+\frac{b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{A b-a B}{a b x^{7/2} (a+b x)} \]

[Out]

-(9*A*b - 7*a*B)/(7*a^2*b*x^(7/2)) + (9*A*b - 7*a*B)/(5*a^3*x^(5/2)) - (b*(9*A*b - 7*a*B))/(3*a^4*x^(3/2)) + (
b^2*(9*A*b - 7*a*B))/(a^5*Sqrt[x]) + (A*b - a*B)/(a*b*x^(7/2)*(a + b*x)) + (b^(5/2)*(9*A*b - 7*a*B)*ArcTan[(Sq
rt[b]*Sqrt[x])/Sqrt[a]])/a^(11/2)

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Rubi [A]  time = 0.0791465, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {27, 78, 51, 63, 205} \[ \frac{b^2 (9 A b-7 a B)}{a^5 \sqrt{x}}+\frac{b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}-\frac{b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{A b-a B}{a b x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^(9/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

-(9*A*b - 7*a*B)/(7*a^2*b*x^(7/2)) + (9*A*b - 7*a*B)/(5*a^3*x^(5/2)) - (b*(9*A*b - 7*a*B))/(3*a^4*x^(3/2)) + (
b^2*(9*A*b - 7*a*B))/(a^5*Sqrt[x]) + (A*b - a*B)/(a*b*x^(7/2)*(a + b*x)) + (b^(5/2)*(9*A*b - 7*a*B)*ArcTan[(Sq
rt[b]*Sqrt[x])/Sqrt[a]])/a^(11/2)

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 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^{9/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{A+B x}{x^{9/2} (a+b x)^2} \, dx\\ &=\frac{A b-a B}{a b x^{7/2} (a+b x)}-\frac{\left (-\frac{9 A b}{2}+\frac{7 a B}{2}\right ) \int \frac{1}{x^{9/2} (a+b x)} \, dx}{a b}\\ &=-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{A b-a B}{a b x^{7/2} (a+b x)}-\frac{(9 A b-7 a B) \int \frac{1}{x^{7/2} (a+b x)} \, dx}{2 a^2}\\ &=-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}+\frac{A b-a B}{a b x^{7/2} (a+b x)}+\frac{(b (9 A b-7 a B)) \int \frac{1}{x^{5/2} (a+b x)} \, dx}{2 a^3}\\ &=-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}-\frac{b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac{A b-a B}{a b x^{7/2} (a+b x)}-\frac{\left (b^2 (9 A b-7 a B)\right ) \int \frac{1}{x^{3/2} (a+b x)} \, dx}{2 a^4}\\ &=-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}-\frac{b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac{b^2 (9 A b-7 a B)}{a^5 \sqrt{x}}+\frac{A b-a B}{a b x^{7/2} (a+b x)}+\frac{\left (b^3 (9 A b-7 a B)\right ) \int \frac{1}{\sqrt{x} (a+b x)} \, dx}{2 a^5}\\ &=-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}-\frac{b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac{b^2 (9 A b-7 a B)}{a^5 \sqrt{x}}+\frac{A b-a B}{a b x^{7/2} (a+b x)}+\frac{\left (b^3 (9 A b-7 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sqrt{x}\right )}{a^5}\\ &=-\frac{9 A b-7 a B}{7 a^2 b x^{7/2}}+\frac{9 A b-7 a B}{5 a^3 x^{5/2}}-\frac{b (9 A b-7 a B)}{3 a^4 x^{3/2}}+\frac{b^2 (9 A b-7 a B)}{a^5 \sqrt{x}}+\frac{A b-a B}{a b x^{7/2} (a+b x)}+\frac{b^{5/2} (9 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{11/2}}\\ \end{align*}

Mathematica [C]  time = 0.0199746, size = 64, normalized size = 0.42 \[ \frac{(a+b x) (7 a B-9 A b) \, _2F_1\left (-\frac{7}{2},1;-\frac{5}{2};-\frac{b x}{a}\right )+7 a (A b-a B)}{7 a^2 b x^{7/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^(9/2)*(a^2 + 2*a*b*x + b^2*x^2)),x]

[Out]

(7*a*(A*b - a*B) + (-9*A*b + 7*a*B)*(a + b*x)*Hypergeometric2F1[-7/2, 1, -5/2, -((b*x)/a)])/(7*a^2*b*x^(7/2)*(
a + b*x))

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Maple [A]  time = 0.022, size = 163, normalized size = 1.1 \begin{align*} -{\frac{2\,A}{7\,{a}^{2}}{x}^{-{\frac{7}{2}}}}+{\frac{4\,Ab}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}-{\frac{2\,B}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}-2\,{\frac{A{b}^{2}}{{a}^{4}{x}^{3/2}}}+{\frac{4\,bB}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}+8\,{\frac{A{b}^{3}}{{a}^{5}\sqrt{x}}}-6\,{\frac{{b}^{2}B}{{a}^{4}\sqrt{x}}}+{\frac{{b}^{4}A}{{a}^{5} \left ( bx+a \right ) }\sqrt{x}}-{\frac{{b}^{3}B}{{a}^{4} \left ( bx+a \right ) }\sqrt{x}}+9\,{\frac{{b}^{4}A}{{a}^{5}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-7\,{\frac{{b}^{3}B}{{a}^{4}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^(9/2)/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

-2/7*A/a^2/x^(7/2)+4/5/a^3/x^(5/2)*A*b-2/5/a^2/x^(5/2)*B-2*b^2/a^4/x^(3/2)*A+4/3*b/a^3/x^(3/2)*B+8*b^3/a^5/x^(
1/2)*A-6*b^2/a^4/x^(1/2)*B+1/a^5*b^4*x^(1/2)/(b*x+a)*A-1/a^4*b^3*x^(1/2)/(b*x+a)*B+9/a^5*b^4/(a*b)^(1/2)*arcta
n(x^(1/2)*b/(a*b)^(1/2))*A-7/a^4*b^3/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*B

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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^(9/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60087, size = 813, normalized size = 5.31 \begin{align*} \left [-\frac{105 \,{\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} +{\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4}\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 (30 \, A a^{4} + 105 \,{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \,{\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \,{\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 6 \,{\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt{x}}{210 \,{\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}, \frac{105 \,{\left ({\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{5} +{\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a \sqrt{\frac{b}{a}}}{b \sqrt{x}}\right ) -{\left (30 \, A a^{4} + 105 \,{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} x^{4} + 70 \,{\left (7 \, B a^{2} b^{2} - 9 \, A a b^{3}\right )} x^{3} - 14 \,{\left (7 \, B a^{3} b - 9 \, A a^{2} b^{2}\right )} x^{2} + 6 \,{\left (7 \, B a^{4} - 9 \, A a^{3} b\right )} x\right )} \sqrt{x}}{105 \,{\left (a^{5} b x^{5} + a^{6} x^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

[-1/210*(105*((7*B*a*b^3 - 9*A*b^4)*x^5 + (7*B*a^2*b^2 - 9*A*a*b^3)*x^4)*sqrt(-b/a)*log((b*x + 2*a*sqrt(x)*sqr
t(-b/a) - a)/(b*x + a)) + 2*(30*A*a^4 + 105*(7*B*a*b^3 - 9*A*b^4)*x^4 + 70*(7*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 14*
(7*B*a^3*b - 9*A*a^2*b^2)*x^2 + 6*(7*B*a^4 - 9*A*a^3*b)*x)*sqrt(x))/(a^5*b*x^5 + a^6*x^4), 1/105*(105*((7*B*a*
b^3 - 9*A*b^4)*x^5 + (7*B*a^2*b^2 - 9*A*a*b^3)*x^4)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sqrt(x))) - (30*A*a^4 + 10
5*(7*B*a*b^3 - 9*A*b^4)*x^4 + 70*(7*B*a^2*b^2 - 9*A*a*b^3)*x^3 - 14*(7*B*a^3*b - 9*A*a^2*b^2)*x^2 + 6*(7*B*a^4
- 9*A*a^3*b)*x)*sqrt(x))/(a^5*b*x^5 + a^6*x^4)]

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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**(9/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Timed out

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Giac [A]  time = 1.13483, size = 184, normalized size = 1.2 \begin{align*} -\frac{{\left (7 \, B a b^{3} - 9 \, A b^{4}\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} a^{5}} - \frac{B a b^{3} \sqrt{x} - A b^{4} \sqrt{x}}{{\left (b x + a\right )} a^{5}} - \frac{2 \,{\left (315 \, B a b^{2} x^{3} - 420 \, A b^{3} x^{3} - 70 \, B a^{2} b x^{2} + 105 \, A a b^{2} x^{2} + 21 \, B a^{3} x - 42 \, A a^{2} b x + 15 \, A a^{3}\right )}}{105 \, a^{5} x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^(9/2)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

-(7*B*a*b^3 - 9*A*b^4)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^5) - (B*a*b^3*sqrt(x) - A*b^4*sqrt(x))/((b*x +
a)*a^5) - 2/105*(315*B*a*b^2*x^3 - 420*A*b^3*x^3 - 70*B*a^2*b*x^2 + 105*A*a*b^2*x^2 + 21*B*a^3*x - 42*A*a^2*b
*x + 15*A*a^3)/(a^5*x^(7/2))