3.3.8 \(\int \frac {x^5 (c+d x^2)^2}{a+b x^2} \, dx\)

Optimal. Leaf size=103 \[ \frac {a^2 (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}-\frac {a x^2 (b c-a d)^2}{2 b^4}+\frac {x^4 (b c-a d)^2}{4 b^3}+\frac {d x^6 (2 b c-a d)}{6 b^2}+\frac {d^2 x^8}{8 b} \]

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 88} \begin {gather*} \frac {a^2 (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^5}+\frac {d x^6 (2 b c-a d)}{6 b^2}+\frac {x^4 (b c-a d)^2}{4 b^3}-\frac {a x^2 (b c-a d)^2}{2 b^4}+\frac {d^2 x^8}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 116, normalized size = 1.13 \begin {gather*} \frac {\left (a^4 d^2-2 a^3 b c d+a^2 b^2 c^2\right ) \log \left (a+b x^2\right )}{2 b^5}-\frac {a x^2 (a d-b c)^2}{2 b^4}+\frac {x^4 (b c-a d)^2}{4 b^3}+\frac {d x^6 (2 b c-a d)}{6 b^2}+\frac {d^2 x^8}{8 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*(-(b*c) + a*d)^2*x^2)/b^4 + ((b*c - a*d)^2*x^4)/(4*b^3) + (d*(2*b*c - a*d)*x^6)/(6*b^2) + (d^2*x^8)/(8
*b) + ((a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*Log[a + b*x^2])/(2*b^5)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5 \left (c+d x^2\right )^2}{a+b x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.79, size = 138, normalized size = 1.34 \begin {gather*} \frac {3 \, b^{4} d^{2} x^{8} + 4 \, {\left (2 \, b^{4} c d - a b^{3} d^{2}\right )} x^{6} + 6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{4} - 12 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{2} + 12 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x^{2} + a\right )}{24 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*b^4*d^2*x^8 + 4*(2*b^4*c*d - a*b^3*d^2)*x^6 + 6*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^4 - 12*(a*b^3*
c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*x^2 + 12*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(b*x^2 + a))/b^5

________________________________________________________________________________________

giac [A]  time = 0.32, size = 148, normalized size = 1.44 \begin {gather*} \frac {3 \, b^{3} d^{2} x^{8} + 8 \, b^{3} c d x^{6} - 4 \, a b^{2} d^{2} x^{6} + 6 \, b^{3} c^{2} x^{4} - 12 \, a b^{2} c d x^{4} + 6 \, a^{2} b d^{2} x^{4} - 12 \, a b^{2} c^{2} x^{2} + 24 \, a^{2} b c d x^{2} - 12 \, a^{3} d^{2} x^{2}}{24 \, b^{4}} + \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*b^3*d^2*x^8 + 8*b^3*c*d*x^6 - 4*a*b^2*d^2*x^6 + 6*b^3*c^2*x^4 - 12*a*b^2*c*d*x^4 + 6*a^2*b*d^2*x^4 - 1
2*a*b^2*c^2*x^2 + 24*a^2*b*c*d*x^2 - 12*a^3*d^2*x^2)/b^4 + 1/2*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(abs(b
*x^2 + a))/b^5

________________________________________________________________________________________

maple [A]  time = 0.00, size = 165, normalized size = 1.60 \begin {gather*} \frac {d^{2} x^{8}}{8 b}-\frac {a \,d^{2} x^{6}}{6 b^{2}}+\frac {c d \,x^{6}}{3 b}+\frac {a^{2} d^{2} x^{4}}{4 b^{3}}-\frac {a c d \,x^{4}}{2 b^{2}}+\frac {c^{2} x^{4}}{4 b}-\frac {a^{3} d^{2} x^{2}}{2 b^{4}}+\frac {a^{2} c d \,x^{2}}{b^{3}}-\frac {a \,c^{2} x^{2}}{2 b^{2}}+\frac {a^{4} d^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{5}}-\frac {a^{3} c d \ln \left (b \,x^{2}+a \right )}{b^{4}}+\frac {a^{2} c^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.11, size = 137, normalized size = 1.33 \begin {gather*} \frac {3 \, b^{3} d^{2} x^{8} + 4 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{6} + 6 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{4} - 12 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2}}{24 \, b^{4}} + \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*b^3*d^2*x^8 + 4*(2*b^3*c*d - a*b^2*d^2)*x^6 + 6*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x^4 - 12*(a*b^2*c^
2 - 2*a^2*b*c*d + a^3*d^2)*x^2)/b^4 + 1/2*(a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*log(b*x^2 + a)/b^5

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 146, normalized size = 1.42 \begin {gather*} x^4\,\left (\frac {c^2}{4\,b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{4\,b}\right )-x^6\,\left (\frac {a\,d^2}{6\,b^2}-\frac {c\,d}{3\,b}\right )+\frac {\ln \left (b\,x^2+a\right )\,\left (a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2\right )}{2\,b^5}+\frac {d^2\,x^8}{8\,b}-\frac {a\,x^2\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )}{2\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*(c^2/(4*b) + (a*((a*d^2)/b^2 - (2*c*d)/b))/(4*b)) - x^6*((a*d^2)/(6*b^2) - (c*d)/(3*b)) + (log(a + b*x^2)*
(a^4*d^2 + a^2*b^2*c^2 - 2*a^3*b*c*d))/(2*b^5) + (d^2*x^8)/(8*b) - (a*x^2*(c^2/b + (a*((a*d^2)/b^2 - (2*c*d)/b
))/b))/(2*b)

________________________________________________________________________________________

sympy [A]  time = 0.46, size = 122, normalized size = 1.18 \begin {gather*} \frac {a^{2} \left (a d - b c\right )^{2} \log {\left (a + b x^{2} \right )}}{2 b^{5}} + x^{6} \left (- \frac {a d^{2}}{6 b^{2}} + \frac {c d}{3 b}\right ) + x^{4} \left (\frac {a^{2} d^{2}}{4 b^{3}} - \frac {a c d}{2 b^{2}} + \frac {c^{2}}{4 b}\right ) + x^{2} \left (- \frac {a^{3} d^{2}}{2 b^{4}} + \frac {a^{2} c d}{b^{3}} - \frac {a c^{2}}{2 b^{2}}\right ) + \frac {d^{2} x^{8}}{8 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*(a*d - b*c)**2*log(a + b*x**2)/(2*b**5) + x**6*(-a*d**2/(6*b**2) + c*d/(3*b)) + x**4*(a**2*d**2/(4*b**3)
- a*c*d/(2*b**2) + c**2/(4*b)) + x**2*(-a**3*d**2/(2*b**4) + a**2*c*d/b**3 - a*c**2/(2*b**2)) + d**2*x**8/(8*b
)

________________________________________________________________________________________