3.39 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^7} \, dx\)

Optimal. Leaf size=114 \[ 10 a^2 b^2 \log (x) (a B+A b)-\frac{5 a^3 b (a B+2 A b)}{2 x^2}-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{a^5 A}{6 x^6}+\frac{1}{4} b^4 x^4 (5 a B+A b)+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{6} b^5 B x^6 \]

[Out]

-(a^5*A)/(6*x^6) - (a^4*(5*A*b + a*B))/(4*x^4) - (5*a^3*b*(2*A*b + a*B))/(2*x^2) + (5*a*b^3*(A*b + 2*a*B)*x^2)
/2 + (b^4*(A*b + 5*a*B)*x^4)/4 + (b^5*B*x^6)/6 + 10*a^2*b^2*(A*b + a*B)*Log[x]

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Rubi [A]  time = 0.10056, antiderivative size = 114, 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} \[ 10 a^2 b^2 \log (x) (a B+A b)-\frac{5 a^3 b (a B+2 A b)}{2 x^2}-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{a^5 A}{6 x^6}+\frac{1}{4} b^4 x^4 (5 a B+A b)+\frac{5}{2} a b^3 x^2 (2 a B+A b)+\frac{1}{6} b^5 B x^6 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^7,x]

[Out]

-(a^5*A)/(6*x^6) - (a^4*(5*A*b + a*B))/(4*x^4) - (5*a^3*b*(2*A*b + a*B))/(2*x^2) + (5*a*b^3*(A*b + 2*a*B)*x^2)
/2 + (b^4*(A*b + 5*a*B)*x^4)/4 + (b^5*B*x^6)/6 + 10*a^2*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^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5 (A+B x)}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (5 a b^3 (A b+2 a B)+\frac{a^5 A}{x^4}+\frac{a^4 (5 A b+a B)}{x^3}+\frac{5 a^3 b (2 A b+a B)}{x^2}+\frac{10 a^2 b^2 (A b+a B)}{x}+b^4 (A b+5 a B) x+b^5 B x^2\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5 A}{6 x^6}-\frac{a^4 (5 A b+a B)}{4 x^4}-\frac{5 a^3 b (2 A b+a B)}{2 x^2}+\frac{5}{2} a b^3 (A b+2 a B) x^2+\frac{1}{4} b^4 (A b+5 a B) x^4+\frac{1}{6} b^5 B x^6+10 a^2 b^2 (A b+a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0378388, size = 116, normalized size = 1.02 \[ \frac{1}{12} \left (120 a^2 b^2 \log (x) (a B+A b)-\frac{60 a^3 A b^2}{x^2}-\frac{15 a^4 b \left (A+2 B x^2\right )}{x^4}-\frac{a^5 \left (2 A+3 B x^2\right )}{x^6}+60 a^2 b^3 B x^2+15 a b^4 x^2 \left (2 A+B x^2\right )+b^5 x^4 \left (3 A+2 B x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^7,x]

[Out]

((-60*a^3*A*b^2)/x^2 + 60*a^2*b^3*B*x^2 + 15*a*b^4*x^2*(2*A + B*x^2) - (15*a^4*b*(A + 2*B*x^2))/x^4 + b^5*x^4*
(3*A + 2*B*x^2) - (a^5*(2*A + 3*B*x^2))/x^6 + 120*a^2*b^2*(A*b + a*B)*Log[x])/12

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^7,x)

[Out]

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

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Maxima [A]  time = 1.00007, size = 166, normalized size = 1.46 \begin{align*} \frac{1}{6} \, B b^{5} x^{6} + \frac{1}{4} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{4} + \frac{5}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{2} + 5 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left (x^{2}\right ) - \frac{2 \, A a^{5} + 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.42026, size = 269, normalized size = 2.36 \begin{align*} \frac{2 \, B b^{5} x^{12} + 3 \,{\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} \log \left (x\right ) - 2 \, A a^{5} - 30 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 3 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^7,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.52849, size = 124, normalized size = 1.09 \begin{align*} \frac{B b^{5} x^{6}}{6} + 10 a^{2} b^{2} \left (A b + B a\right ) \log{\left (x \right )} + x^{4} \left (\frac{A b^{5}}{4} + \frac{5 B a b^{4}}{4}\right ) + x^{2} \left (\frac{5 A a b^{4}}{2} + 5 B a^{2} b^{3}\right ) - \frac{2 A a^{5} + x^{4} \left (60 A a^{3} b^{2} + 30 B a^{4} b\right ) + x^{2} \left (15 A a^{4} b + 3 B a^{5}\right )}{12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**7,x)

[Out]

B*b**5*x**6/6 + 10*a**2*b**2*(A*b + B*a)*log(x) + x**4*(A*b**5/4 + 5*B*a*b**4/4) + x**2*(5*A*a*b**4/2 + 5*B*a*
*2*b**3) - (2*A*a**5 + x**4*(60*A*a**3*b**2 + 30*B*a**4*b) + x**2*(15*A*a**4*b + 3*B*a**5))/(12*x**6)

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Giac [A]  time = 1.44273, size = 204, normalized size = 1.79 \begin{align*} \frac{1}{6} \, B b^{5} x^{6} + \frac{5}{4} \, B a b^{4} x^{4} + \frac{1}{4} \, A b^{5} x^{4} + 5 \, B a^{2} b^{3} x^{2} + \frac{5}{2} \, A a b^{4} x^{2} + 5 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} \log \left (x^{2}\right ) - \frac{110 \, B a^{3} b^{2} x^{6} + 110 \, A a^{2} b^{3} x^{6} + 30 \, B a^{4} b x^{4} + 60 \, A a^{3} b^{2} x^{4} + 3 \, B a^{5} x^{2} + 15 \, A a^{4} b x^{2} + 2 \, A a^{5}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^7,x, algorithm="giac")

[Out]

1/6*B*b^5*x^6 + 5/4*B*a*b^4*x^4 + 1/4*A*b^5*x^4 + 5*B*a^2*b^3*x^2 + 5/2*A*a*b^4*x^2 + 5*(B*a^3*b^2 + A*a^2*b^3
)*log(x^2) - 1/12*(110*B*a^3*b^2*x^6 + 110*A*a^2*b^3*x^6 + 30*B*a^4*b*x^4 + 60*A*a^3*b^2*x^4 + 3*B*a^5*x^2 + 1
5*A*a^4*b*x^2 + 2*A*a^5)/x^6