3.218 \(\int \frac{x^5 (c+d x^2)^3}{a+b x^2} \, dx\)

Optimal. Leaf size=138 \[ \frac{d x^6 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{6 b^3}+\frac{a^2 (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^6}+\frac{d^2 x^8 (3 b c-a d)}{8 b^2}+\frac{x^4 (b c-a d)^3}{4 b^4}-\frac{a x^2 (b c-a d)^3}{2 b^5}+\frac{d^3 x^{10}}{10 b} \]

[Out]

-(a*(b*c - a*d)^3*x^2)/(2*b^5) + ((b*c - a*d)^3*x^4)/(4*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^6)/(6*b^
3) + (d^2*(3*b*c - a*d)*x^8)/(8*b^2) + (d^3*x^10)/(10*b) + (a^2*(b*c - a*d)^3*Log[a + b*x^2])/(2*b^6)

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Rubi [A]  time = 0.178359, antiderivative size = 138, normalized size of antiderivative = 1., 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} \[ \frac{d x^6 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{6 b^3}+\frac{a^2 (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^6}+\frac{d^2 x^8 (3 b c-a d)}{8 b^2}+\frac{x^4 (b c-a d)^3}{4 b^4}-\frac{a x^2 (b c-a d)^3}{2 b^5}+\frac{d^3 x^{10}}{10 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

-(a*(b*c - a*d)^3*x^2)/(2*b^5) + ((b*c - a*d)^3*x^4)/(4*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^6)/(6*b^
3) + (d^2*(3*b*c - a*d)*x^8)/(8*b^2) + (d^3*x^10)/(10*b) + (a^2*(b*c - a*d)^3*Log[a + b*x^2])/(2*b^6)

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 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]))

Rubi steps

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

Mathematica [A]  time = 0.0765649, size = 128, normalized size = 0.93 \[ \frac{20 b^3 d x^6 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )+60 a^2 (b c-a d)^3 \log \left (a+b x^2\right )+15 b^4 d^2 x^8 (3 b c-a d)+30 b^2 x^4 (b c-a d)^3+60 a b x^2 (a d-b c)^3+12 b^5 d^3 x^{10}}{120 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(c + d*x^2)^3)/(a + b*x^2),x]

[Out]

(60*a*b*(-(b*c) + a*d)^3*x^2 + 30*b^2*(b*c - a*d)^3*x^4 + 20*b^3*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^6 + 15*
b^4*d^2*(3*b*c - a*d)*x^8 + 12*b^5*d^3*x^10 + 60*a^2*(b*c - a*d)^3*Log[a + b*x^2])/(120*b^6)

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Maple [B]  time = 0.005, size = 263, normalized size = 1.9 \begin{align*}{\frac{{d}^{3}{x}^{10}}{10\,b}}-{\frac{{x}^{8}a{d}^{3}}{8\,{b}^{2}}}+{\frac{3\,{x}^{8}c{d}^{2}}{8\,b}}+{\frac{{x}^{6}{a}^{2}{d}^{3}}{6\,{b}^{3}}}-{\frac{{x}^{6}ac{d}^{2}}{2\,{b}^{2}}}+{\frac{{x}^{6}{c}^{2}d}{2\,b}}-{\frac{{x}^{4}{a}^{3}{d}^{3}}{4\,{b}^{4}}}+{\frac{3\,{x}^{4}{a}^{2}c{d}^{2}}{4\,{b}^{3}}}-{\frac{3\,{x}^{4}a{c}^{2}d}{4\,{b}^{2}}}+{\frac{{x}^{4}{c}^{3}}{4\,b}}+{\frac{{a}^{4}{d}^{3}{x}^{2}}{2\,{b}^{5}}}-{\frac{3\,{a}^{3}c{d}^{2}{x}^{2}}{2\,{b}^{4}}}+{\frac{3\,{x}^{2}{a}^{2}{c}^{2}d}{2\,{b}^{3}}}-{\frac{a{c}^{3}{x}^{2}}{2\,{b}^{2}}}-{\frac{{a}^{5}\ln \left ( b{x}^{2}+a \right ){d}^{3}}{2\,{b}^{6}}}+{\frac{3\,{a}^{4}\ln \left ( b{x}^{2}+a \right ) c{d}^{2}}{2\,{b}^{5}}}-{\frac{3\,{a}^{3}\ln \left ( b{x}^{2}+a \right ){c}^{2}d}{2\,{b}^{4}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(d*x^2+c)^3/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.0417, size = 296, normalized size = 2.14 \begin{align*} \frac{12 \, b^{4} d^{3} x^{10} + 15 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{8} + 20 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{6} + 30 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} - 60 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2}}{120 \, b^{5}} + \frac{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d*x^2+c)^3/(b*x^2+a),x, algorithm="maxima")

[Out]

1/120*(12*b^4*d^3*x^10 + 15*(3*b^4*c*d^2 - a*b^3*d^3)*x^8 + 20*(3*b^4*c^2*d - 3*a*b^3*c*d^2 + a^2*b^2*d^3)*x^6
+ 30*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*x^4 - 60*(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*
c*d^2 - a^4*d^3)*x^2)/b^5 + 1/2*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*log(b*x^2 + a)/b^6

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Fricas [A]  time = 1.45105, size = 447, normalized size = 3.24 \begin{align*} \frac{12 \, b^{5} d^{3} x^{10} + 15 \,{\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{8} + 20 \,{\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{6} + 30 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{4} - 60 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} + 60 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x^{2} + a\right )}{120 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d*x^2+c)^3/(b*x^2+a),x, algorithm="fricas")

[Out]

1/120*(12*b^5*d^3*x^10 + 15*(3*b^5*c*d^2 - a*b^4*d^3)*x^8 + 20*(3*b^5*c^2*d - 3*a*b^4*c*d^2 + a^2*b^3*d^3)*x^6
+ 30*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^4 - 60*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*
b^2*c*d^2 - a^4*b*d^3)*x^2 + 60*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*log(b*x^2 + a))/b^6

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Sympy [A]  time = 0.729092, size = 187, normalized size = 1.36 \begin{align*} - \frac{a^{2} \left (a d - b c\right )^{3} \log{\left (a + b x^{2} \right )}}{2 b^{6}} + \frac{d^{3} x^{10}}{10 b} - \frac{x^{8} \left (a d^{3} - 3 b c d^{2}\right )}{8 b^{2}} + \frac{x^{6} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{6 b^{3}} - \frac{x^{4} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{4 b^{4}} + \frac{x^{2} \left (a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}\right )}{2 b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(d*x**2+c)**3/(b*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.15839, size = 321, normalized size = 2.33 \begin{align*} \frac{12 \, b^{4} d^{3} x^{10} + 45 \, b^{4} c d^{2} x^{8} - 15 \, a b^{3} d^{3} x^{8} + 60 \, b^{4} c^{2} d x^{6} - 60 \, a b^{3} c d^{2} x^{6} + 20 \, a^{2} b^{2} d^{3} x^{6} + 30 \, b^{4} c^{3} x^{4} - 90 \, a b^{3} c^{2} d x^{4} + 90 \, a^{2} b^{2} c d^{2} x^{4} - 30 \, a^{3} b d^{3} x^{4} - 60 \, a b^{3} c^{3} x^{2} + 180 \, a^{2} b^{2} c^{2} d x^{2} - 180 \, a^{3} b c d^{2} x^{2} + 60 \, a^{4} d^{3} x^{2}}{120 \, b^{5}} + \frac{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(d*x^2+c)^3/(b*x^2+a),x, algorithm="giac")

[Out]

1/120*(12*b^4*d^3*x^10 + 45*b^4*c*d^2*x^8 - 15*a*b^3*d^3*x^8 + 60*b^4*c^2*d*x^6 - 60*a*b^3*c*d^2*x^6 + 20*a^2*
b^2*d^3*x^6 + 30*b^4*c^3*x^4 - 90*a*b^3*c^2*d*x^4 + 90*a^2*b^2*c*d^2*x^4 - 30*a^3*b*d^3*x^4 - 60*a*b^3*c^3*x^2
+ 180*a^2*b^2*c^2*d*x^2 - 180*a^3*b*c*d^2*x^2 + 60*a^4*d^3*x^2)/b^5 + 1/2*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*
a^4*b*c*d^2 - a^5*d^3)*log(abs(b*x^2 + a))/b^6