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

Optimal. Leaf size=130 \[ \frac{a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}-\frac{x^{5/2} (5 A b-7 a B)}{5 a b^2}+\frac{x^{3/2} (5 A b-7 a B)}{3 b^3}-\frac{a \sqrt{x} (5 A b-7 a B)}{b^4}+\frac{x^{7/2} (A b-a B)}{a b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0658589, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {27, 78, 50, 63, 205} \[ \frac{a^{3/2} (5 A b-7 a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}}-\frac{x^{5/2} (5 A b-7 a B)}{5 a b^2}+\frac{x^{3/2} (5 A b-7 a B)}{3 b^3}-\frac{a \sqrt{x} (5 A b-7 a B)}{b^4}+\frac{x^{7/2} (A b-a B)}{a b (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0755441, size = 110, normalized size = 0.85 \[ \frac{\sqrt{x} \left (a^2 (70 b B x-75 A b)+105 a^3 B-2 a b^2 x (25 A+7 B x)+2 b^3 x^2 (5 A+3 B x)\right )}{15 b^4 (a+b x)}-\frac{a^{3/2} (7 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(105*a^3*B + 2*b^3*x^2*(5*A + 3*B*x) - 2*a*b^2*x*(25*A + 7*B*x) + a^2*(-75*A*b + 70*b*B*x)))/(15*b^4*
(a + b*x)) - (a^(3/2)*(-5*A*b + 7*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(9/2)

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Maple [A]  time = 0.014, size = 139, normalized size = 1.1 \begin{align*}{\frac{2\,B}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}+{\frac{2\,A}{3\,{b}^{2}}{x}^{{\frac{3}{2}}}}-{\frac{4\,aB}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-4\,{\frac{aA\sqrt{x}}{{b}^{3}}}+6\,{\frac{{a}^{2}B\sqrt{x}}{{b}^{4}}}-{\frac{A{a}^{2}}{{b}^{3} \left ( bx+a \right ) }\sqrt{x}}+{\frac{B{a}^{3}}{{b}^{4} \left ( bx+a \right ) }\sqrt{x}}+5\,{\frac{A{a}^{2}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{\sqrt{x}b}{\sqrt{ab}}} \right ) }-7\,{\frac{B{a}^{3}}{{b}^{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(x^(5/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

2/5/b^2*B*x^(5/2)+2/3/b^2*A*x^(3/2)-4/3/b^3*B*x^(3/2)*a-4/b^3*A*a*x^(1/2)+6/b^4*B*a^2*x^(1/2)-a^2/b^3*x^(1/2)/
(b*x+a)*A+a^3/b^4*x^(1/2)/(b*x+a)*B+5*a^2/b^3/(a*b)^(1/2)*arctan(x^(1/2)*b/(a*b)^(1/2))*A-7*a^3/b^4/(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(x^(5/2)*(B*x+A)/(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58304, size = 652, normalized size = 5.02 \begin{align*} \left [-\frac{15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b +{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \,{\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{x}}{30 \,{\left (b^{5} x + a b^{4}\right )}}, -\frac{15 \,{\left (7 \, B a^{3} - 5 \, A a^{2} b +{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (6 \, B b^{3} x^{3} + 105 \, B a^{3} - 75 \, A a^{2} b - 2 \,{\left (7 \, B a b^{2} - 5 \, A b^{3}\right )} x^{2} + 10 \,{\left (7 \, B a^{2} b - 5 \, A a b^{2}\right )} x\right )} \sqrt{x}}{15 \,{\left (b^{5} x + a b^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/30*(15*(7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)
/(b*x + a)) - 2*(6*B*b^3*x^3 + 105*B*a^3 - 75*A*a^2*b - 2*(7*B*a*b^2 - 5*A*b^3)*x^2 + 10*(7*B*a^2*b - 5*A*a*b^
2)*x)*sqrt(x))/(b^5*x + a*b^4), -1/15*(15*(7*B*a^3 - 5*A*a^2*b + (7*B*a^2*b - 5*A*a*b^2)*x)*sqrt(a/b)*arctan(b
*sqrt(x)*sqrt(a/b)/a) - (6*B*b^3*x^3 + 105*B*a^3 - 75*A*a^2*b - 2*(7*B*a*b^2 - 5*A*b^3)*x^2 + 10*(7*B*a^2*b -
5*A*a*b^2)*x)*sqrt(x))/(b^5*x + a*b^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(B*x+A)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.17089, size = 165, normalized size = 1.27 \begin{align*} -\frac{{\left (7 \, B a^{3} - 5 \, A a^{2} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{4}} + \frac{B a^{3} \sqrt{x} - A a^{2} b \sqrt{x}}{{\left (b x + a\right )} b^{4}} + \frac{2 \,{\left (3 \, B b^{8} x^{\frac{5}{2}} - 10 \, B a b^{7} x^{\frac{3}{2}} + 5 \, A b^{8} x^{\frac{3}{2}} + 45 \, B a^{2} b^{6} \sqrt{x} - 30 \, A a b^{7} \sqrt{x}\right )}}{15 \, b^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(7*B*a^3 - 5*A*a^2*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^4) + (B*a^3*sqrt(x) - A*a^2*b*sqrt(x))/((b*x +
a)*b^4) + 2/15*(3*B*b^8*x^(5/2) - 10*B*a*b^7*x^(3/2) + 5*A*b^8*x^(3/2) + 45*B*a^2*b^6*sqrt(x) - 30*A*a*b^7*sq
rt(x))/b^10