3.1.37 \(\int \frac {(a+b x^2)^5 (A+B x^2)}{x^5} \, dx\) [37]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77} \begin {gather*} -\frac {a^5 A}{4 x^4}-\frac {a^4 (a B+5 A b)}{2 x^2}+5 a^3 b \log (x) (a B+2 A b)+5 a^2 b^2 x^2 (a B+A b)+\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 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a^5*A)/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 77

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])

Rule 457

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]]

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} \text {Subst}\left (\int \frac {(a+b x)^5 (A+B x)}{x^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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*}

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Mathematica [A]
time = 0.03, size = 112, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(a^5*A)/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 = 0.07, size = 117, normalized size = 1.04

method	result	size
default	\(\frac {b^{5} B \,x^{8}}{8}+\frac {A \,b^{5} x^{6}}{6}+\frac {5 B a \,b^{4} x^{6}}{6}+\frac {5 A a \,b^{4} x^{4}}{4}+\frac {5 B \,a^{2} b^{3} x^{4}}{2}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}-\frac {a^{5} A}{4 x^{4}}-\frac {a^{4} \left (5 A b +B a \right )}{2 x^{2}}+5 a^{3} b \left (2 A b +B a \right ) \ln \left (x \right )\)	\(117\)
norman	\(\frac {\left (\frac {1}{6} b^{5} A +\frac {5}{6} a \,b^{4} B \right ) x^{10}+\left (\frac {5}{4} a \,b^{4} A +\frac {5}{2} a^{2} b^{3} B \right ) x^{8}+\left (-\frac {5}{2} a^{4} b A -\frac {1}{2} a^{5} B \right ) x^{2}+\left (5 a^{2} b^{3} A +5 a^{3} b^{2} B \right ) x^{6}-\frac {a^{5} A}{4}+\frac {b^{5} B \,x^{12}}{8}}{x^{4}}+\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) \ln \left (x \right )\)	\(122\)
risch	\(\frac {b^{5} B \,x^{8}}{8}+\frac {A \,b^{5} x^{6}}{6}+\frac {5 B a \,b^{4} x^{6}}{6}+\frac {5 A a \,b^{4} x^{4}}{4}+\frac {5 B \,a^{2} b^{3} x^{4}}{2}+5 A \,a^{2} b^{3} x^{2}+5 B \,a^{3} b^{2} x^{2}+\frac {\left (-\frac {5}{2} a^{4} b A -\frac {1}{2} a^{5} B \right ) x^{2}-\frac {a^{5} A}{4}}{x^{4}}+10 A \ln \left (x \right ) a^{3} b^{2}+5 B \ln \left (x \right ) a^{4} b\)	\(125\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 122, normalized size = 1.09 \begin {gather*} \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 {gather*}

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 = 0.71, size = 123, normalized size = 1.10 \begin {gather*} \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 {gather*}

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 = 0.33, size = 128, normalized size = 1.14 \begin {gather*} \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} \cdot \left (\frac {5 A a b^{4}}{4} + \frac {5 B a^{2} b^{3}}{2}\right ) + x^{2} \cdot \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 {gather*}

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 = 0.70, size = 149, normalized size = 1.33 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.03, size = 113, normalized size = 1.01 \begin {gather*} \ln \left (x\right )\,\left (5\,B\,a^4\,b+10\,A\,a^3\,b^2\right )-\frac {\frac {A\,a^5}{4}+x^2\,\left (\frac {B\,a^5}{2}+\frac {5\,A\,b\,a^4}{2}\right )}{x^4}+x^6\,\left (\frac {A\,b^5}{6}+\frac {5\,B\,a\,b^4}{6}\right )+\frac {B\,b^5\,x^8}{8}+5\,a^2\,b^2\,x^2\,\left (A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^4\,\left (A\,b+2\,B\,a\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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