3.2.98 \(\int \frac {x^5 (c+d x^2)}{a+b x^2} \, dx\) [198]

Optimal. Leaf size=75 \[ -\frac {a (b c-a d) x^2}{2 b^3}+\frac {(b c-a d) x^4}{4 b^2}+\frac {d x^6}{6 b}+\frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4} \]

[Out]

-1/2*a*(-a*d+b*c)*x^2/b^3+1/4*(-a*d+b*c)*x^4/b^2+1/6*d*x^6/b+1/2*a^2*(-a*d+b*c)*ln(b*x^2+a)/b^4

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Rubi [A]
time = 0.06, antiderivative size = 75, 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, 78} \begin {gather*} \frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}-\frac {a x^2 (b c-a d)}{2 b^3}+\frac {x^4 (b c-a d)}{4 b^2}+\frac {d x^6}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(c + d*x^2))/(a + b*x^2),x]

[Out]

-1/2*(a*(b*c - a*d)*x^2)/b^3 + ((b*c - a*d)*x^4)/(4*b^2) + (d*x^6)/(6*b) + (a^2*(b*c - a*d)*Log[a + b*x^2])/(2
*b^4)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 {x^5 \left (c+d x^2\right )}{a+b x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2 (c+d x)}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {a (-b c+a d)}{b^3}+\frac {(b c-a d) x}{b^2}+\frac {d x^2}{b}-\frac {a^2 (-b c+a d)}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {a (b c-a d) x^2}{2 b^3}+\frac {(b c-a d) x^4}{4 b^2}+\frac {d x^6}{6 b}+\frac {a^2 (b c-a d) \log \left (a+b x^2\right )}{2 b^4}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 71, normalized size = 0.95 \begin {gather*} \frac {b x^2 \left (6 a^2 d-3 a b \left (2 c+d x^2\right )+b^2 x^2 \left (3 c+2 d x^2\right )\right )+6 a^2 (b c-a d) \log \left (a+b x^2\right )}{12 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(c + d*x^2))/(a + b*x^2),x]

[Out]

(b*x^2*(6*a^2*d - 3*a*b*(2*c + d*x^2) + b^2*x^2*(3*c + 2*d*x^2)) + 6*a^2*(b*c - a*d)*Log[a + b*x^2])/(12*b^4)

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Maple [A]
time = 0.07, size = 74, normalized size = 0.99

method	result	size
norman	\(-\frac {\left (a d -b c \right ) x^{4}}{4 b^{2}}+\frac {d \,x^{6}}{6 b}+\frac {a \left (a d -b c \right ) x^{2}}{2 b^{3}}-\frac {a^{2} \left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\)	\(68\)
default	\(\frac {\frac {1}{3} b^{2} d \,x^{6}-\frac {1}{2} a b d \,x^{4}+\frac {1}{2} b^{2} c \,x^{4}+a^{2} d \,x^{2}-a b c \,x^{2}}{2 b^{3}}-\frac {a^{2} \left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{4}}\)	\(74\)
risch	\(\frac {d \,x^{6}}{6 b}-\frac {a d \,x^{4}}{4 b^{2}}+\frac {c \,x^{4}}{4 b}+\frac {a^{2} d \,x^{2}}{2 b^{3}}-\frac {a c \,x^{2}}{2 b^{2}}-\frac {a^{3} \ln \left (b \,x^{2}+a \right ) d}{2 b^{4}}+\frac {a^{2} \ln \left (b \,x^{2}+a \right ) c}{2 b^{3}}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/b^3*(1/3*b^2*d*x^6-1/2*a*b*d*x^4+1/2*b^2*c*x^4+a^2*d*x^2-a*b*c*x^2)-1/2*a^2*(a*d-b*c)/b^4*ln(b*x^2+a)

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Maxima [A]
time = 0.29, size = 74, normalized size = 0.99 \begin {gather*} \frac {2 \, b^{2} d x^{6} + 3 \, {\left (b^{2} c - a b d\right )} x^{4} - 6 \, {\left (a b c - a^{2} d\right )} x^{2}}{12 \, b^{3}} + \frac {{\left (a^{2} b c - a^{3} d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*b^2*d*x^6 + 3*(b^2*c - a*b*d)*x^4 - 6*(a*b*c - a^2*d)*x^2)/b^3 + 1/2*(a^2*b*c - a^3*d)*log(b*x^2 + a)/
b^4

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Fricas [A]
time = 1.35, size = 75, normalized size = 1.00 \begin {gather*} \frac {2 \, b^{3} d x^{6} + 3 \, {\left (b^{3} c - a b^{2} d\right )} x^{4} - 6 \, {\left (a b^{2} c - a^{2} b d\right )} x^{2} + 6 \, {\left (a^{2} b c - a^{3} d\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*b^3*d*x^6 + 3*(b^3*c - a*b^2*d)*x^4 - 6*(a*b^2*c - a^2*b*d)*x^2 + 6*(a^2*b*c - a^3*d)*log(b*x^2 + a))/
b^4

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Sympy [A]
time = 0.16, size = 70, normalized size = 0.93 \begin {gather*} - \frac {a^{2} \left (a d - b c\right ) \log {\left (a + b x^{2} \right )}}{2 b^{4}} + x^{4} \left (- \frac {a d}{4 b^{2}} + \frac {c}{4 b}\right ) + x^{2} \left (\frac {a^{2} d}{2 b^{3}} - \frac {a c}{2 b^{2}}\right ) + \frac {d x^{6}}{6 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(d*x**2+c)/(b*x**2+a),x)

[Out]

-a**2*(a*d - b*c)*log(a + b*x**2)/(2*b**4) + x**4*(-a*d/(4*b**2) + c/(4*b)) + x**2*(a**2*d/(2*b**3) - a*c/(2*b
**2)) + d*x**6/(6*b)

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Giac [A]
time = 1.14, size = 77, normalized size = 1.03 \begin {gather*} \frac {2 \, b^{2} d x^{6} + 3 \, b^{2} c x^{4} - 3 \, a b d x^{4} - 6 \, a b c x^{2} + 6 \, a^{2} d x^{2}}{12 \, b^{3}} + \frac {{\left (a^{2} b c - a^{3} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*b^2*d*x^6 + 3*b^2*c*x^4 - 3*a*b*d*x^4 - 6*a*b*c*x^2 + 6*a^2*d*x^2)/b^3 + 1/2*(a^2*b*c - a^3*d)*log(abs
(b*x^2 + a))/b^4

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Mupad [B]
time = 0.06, size = 76, normalized size = 1.01 \begin {gather*} x^4\,\left (\frac {c}{4\,b}-\frac {a\,d}{4\,b^2}\right )+\frac {d\,x^6}{6\,b}-\frac {\ln \left (b\,x^2+a\right )\,\left (a^3\,d-a^2\,b\,c\right )}{2\,b^4}-\frac {a\,x^2\,\left (\frac {c}{b}-\frac {a\,d}{b^2}\right )}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(c + d*x^2))/(a + b*x^2),x)

[Out]

x^4*(c/(4*b) - (a*d)/(4*b^2)) + (d*x^6)/(6*b) - (log(a + b*x^2)*(a^3*d - a^2*b*c))/(2*b^4) - (a*x^2*(c/b - (a*
d)/b^2))/(2*b)

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