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

Optimal. Leaf size=112 \[ 5 a^2 b^2 x^2 (a B+A b)-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)-\frac{a^5 A}{4 x^4}+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{8} b^5 B x^8 \]

[Out]

-(a^5*A)/(4*x^4) - (a^4*(5*A*b + a*B))/(2*x^2) + 5*a^2*b^2*(A*b + a*B)*x^2 + (5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (
b^4*(A*b + 5*a*B)*x^6)/6 + (b^5*B*x^8)/8 + 5*a^3*b*(2*A*b + a*B)*Log[x]

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Rubi [A]  time = 0.10434, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ 5 a^2 b^2 x^2 (a B+A b)-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)-\frac{a^5 A}{4 x^4}+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{8} b^5 B x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*A)/(4*x^4) - (a^4*(5*A*b + a*B))/(2*x^2) + 5*a^2*b^2*(A*b + a*B)*x^2 + (5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (
b^4*(A*b + 5*a*B)*x^6)/6 + (b^5*B*x^8)/8 + 5*a^3*b*(2*A*b + a*B)*Log[x]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^5 \left (A+B x^2\right )}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (10 a^2 b^2 (A b+a B)+\frac{a^5 A}{x^3}+\frac{a^4 (5 A b+a B)}{x^2}+\frac{5 a^3 b (2 A b+a B)}{x}+5 a b^3 (A b+2 a B) x+b^4 (A b+5 a B) x^2+b^5 B x^3\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 A}{4 x^4}-\frac{a^4 (5 A b+a B)}{2 x^2}+5 a^2 b^2 (A b+a B) x^2+\frac{5}{4} a b^3 (A b+2 a B) x^4+\frac{1}{6} b^4 (A b+5 a B) x^6+\frac{1}{8} b^5 B x^8+5 a^3 b (2 A b+a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0375762, size = 112, normalized size = 1. \[ 5 a^2 b^2 x^2 (a B+A b)-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)-\frac{a^5 A}{4 x^4}+\frac{1}{6} b^4 x^6 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{8} b^5 B x^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^5*A)/(4*x^4) - (a^4*(5*A*b + a*B))/(2*x^2) + 5*a^2*b^2*(A*b + a*B)*x^2 + (5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (
b^4*(A*b + 5*a*B)*x^6)/6 + (b^5*B*x^8)/8 + 5*a^3*b*(2*A*b + a*B)*Log[x]

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Maple [A]  time = 0.007, size = 124, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{8}}{8}}+{\frac{A{x}^{6}{b}^{5}}{6}}+{\frac{5\,B{x}^{6}a{b}^{4}}{6}}+{\frac{5\,A{x}^{4}a{b}^{4}}{4}}+{\frac{5\,B{x}^{4}{a}^{2}{b}^{3}}{2}}+5\,A{x}^{2}{a}^{2}{b}^{3}+5\,B{x}^{2}{a}^{3}{b}^{2}+10\,A\ln \left ( x \right ){a}^{3}{b}^{2}+5\,B\ln \left ( x \right ){a}^{4}b-{\frac{A{a}^{5}}{4\,{x}^{4}}}-{\frac{5\,{a}^{4}bA}{2\,{x}^{2}}}-{\frac{{a}^{5}B}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*b^5*B*x^8+1/6*A*x^6*b^5+5/6*B*x^6*a*b^4+5/4*A*x^4*a*b^4+5/2*B*x^4*a^2*b^3+5*A*x^2*a^2*b^3+5*B*x^2*a^3*b^2+
10*A*ln(x)*a^3*b^2+5*B*ln(x)*a^4*b-1/4*a^5*A/x^4-5/2*a^4/x^2*A*b-1/2*a^5/x^2*B

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Maxima [A]  time = 1.00494, size = 165, normalized size = 1.47 \begin{align*} \frac{1}{8} \, B b^{5} x^{8} + \frac{1}{6} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 5 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x^{2}\right ) - \frac{A a^{5} + 2 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*b^5*x^8 + 1/6*(5*B*a*b^4 + A*b^5)*x^6 + 5/4*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 5*(B*a^3*b^2 + A*a^2*b^3)*x^2
+ 5/2*(B*a^4*b + 2*A*a^3*b^2)*log(x^2) - 1/4*(A*a^5 + 2*(B*a^5 + 5*A*a^4*b)*x^2)/x^4

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Fricas [A]  time = 1.46823, size = 271, normalized size = 2.42 \begin{align*} \frac{3 \, B b^{5} x^{12} + 4 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 30 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 120 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 6 \, A a^{5} + 120 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} \log \left (x\right ) - 12 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{24 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*B*b^5*x^12 + 4*(5*B*a*b^4 + A*b^5)*x^10 + 30*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 120*(B*a^3*b^2 + A*a^2*b^3)
*x^6 - 6*A*a^5 + 120*(B*a^4*b + 2*A*a^3*b^2)*x^4*log(x) - 12*(B*a^5 + 5*A*a^4*b)*x^2)/x^4

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Sympy [A]  time = 0.794991, size = 126, normalized size = 1.12 \begin{align*} \frac{B b^{5} x^{8}}{8} + 5 a^{3} b \left (2 A b + B a\right ) \log{\left (x \right )} + x^{6} \left (\frac{A b^{5}}{6} + \frac{5 B a b^{4}}{6}\right ) + x^{4} \left (\frac{5 A a b^{4}}{4} + \frac{5 B a^{2} b^{3}}{2}\right ) + x^{2} \left (5 A a^{2} b^{3} + 5 B a^{3} b^{2}\right ) - \frac{A a^{5} + x^{2} \left (10 A a^{4} b + 2 B a^{5}\right )}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b**5*x**8/8 + 5*a**3*b*(2*A*b + B*a)*log(x) + x**6*(A*b**5/6 + 5*B*a*b**4/6) + x**4*(5*A*a*b**4/4 + 5*B*a**2
*b**3/2) + x**2*(5*A*a**2*b**3 + 5*B*a**3*b**2) - (A*a**5 + x**2*(10*A*a**4*b + 2*B*a**5))/(4*x**4)

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Giac [A]  time = 1.14906, size = 201, normalized size = 1.79 \begin{align*} \frac{1}{8} \, B b^{5} x^{8} + \frac{5}{6} \, B a b^{4} x^{6} + \frac{1}{6} \, A b^{5} x^{6} + \frac{5}{2} \, B a^{2} b^{3} x^{4} + \frac{5}{4} \, A a b^{4} x^{4} + 5 \, B a^{3} b^{2} x^{2} + 5 \, A a^{2} b^{3} x^{2} + \frac{5}{2} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} \log \left (x^{2}\right ) - \frac{15 \, B a^{4} b x^{4} + 30 \, A a^{3} b^{2} x^{4} + 2 \, B a^{5} x^{2} + 10 \, A a^{4} b x^{2} + A a^{5}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*B*b^5*x^8 + 5/6*B*a*b^4*x^6 + 1/6*A*b^5*x^6 + 5/2*B*a^2*b^3*x^4 + 5/4*A*a*b^4*x^4 + 5*B*a^3*b^2*x^2 + 5*A*
a^2*b^3*x^2 + 5/2*(B*a^4*b + 2*A*a^3*b^2)*log(x^2) - 1/4*(15*B*a^4*b*x^4 + 30*A*a^3*b^2*x^4 + 2*B*a^5*x^2 + 10
*A*a^4*b*x^2 + A*a^5)/x^4