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

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

[Out]

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

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Rubi [A]  time = 0.18, antiderivative size = 138, 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} \[ \frac {d x^6 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{6 b^3}+\frac {a^2 (b c-a d)^3 \log \left (a+b x^2\right )}{2 b^6}+\frac {d^2 x^8 (3 b c-a d)}{8 b^2}+\frac {x^4 (b c-a d)^3}{4 b^4}-\frac {a x^2 (b c-a d)^3}{2 b^5}+\frac {d^3 x^{10}}{10 b} \]

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 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 )^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*}

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Mathematica [A]  time = 0.07, 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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fricas [A]  time = 0.46, size = 220, normalized size = 1.59 \[ \frac {12 \, b^{5} d^{3} x^{10} + 15 \, {\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{8} + 20 \, {\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{6} + 30 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{4} - 60 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} + 60 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x^{2} + a\right )}{120 \, b^{6}} \]

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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giac [A]  time = 0.32, size = 238, normalized size = 1.72 \[ \frac {12 \, b^{4} d^{3} x^{10} + 45 \, b^{4} c d^{2} x^{8} - 15 \, a b^{3} d^{3} x^{8} + 60 \, b^{4} c^{2} d x^{6} - 60 \, a b^{3} c d^{2} x^{6} + 20 \, a^{2} b^{2} d^{3} x^{6} + 30 \, b^{4} c^{3} x^{4} - 90 \, a b^{3} c^{2} d x^{4} + 90 \, a^{2} b^{2} c d^{2} x^{4} - 30 \, a^{3} b d^{3} x^{4} - 60 \, a b^{3} c^{3} x^{2} + 180 \, a^{2} b^{2} c^{2} d x^{2} - 180 \, a^{3} b c d^{2} x^{2} + 60 \, a^{4} d^{3} x^{2}}{120 \, b^{5}} + \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} \]

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

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maple [B]  time = 0.01, size = 263, normalized size = 1.91 \[ \frac {d^{3} x^{10}}{10 b}-\frac {a \,d^{3} x^{8}}{8 b^{2}}+\frac {3 c \,d^{2} x^{8}}{8 b}+\frac {a^{2} d^{3} x^{6}}{6 b^{3}}-\frac {a c \,d^{2} x^{6}}{2 b^{2}}+\frac {c^{2} d \,x^{6}}{2 b}-\frac {a^{3} d^{3} x^{4}}{4 b^{4}}+\frac {3 a^{2} c \,d^{2} x^{4}}{4 b^{3}}-\frac {3 a \,c^{2} d \,x^{4}}{4 b^{2}}+\frac {c^{3} x^{4}}{4 b}+\frac {a^{4} d^{3} x^{2}}{2 b^{5}}-\frac {3 a^{3} c \,d^{2} x^{2}}{2 b^{4}}+\frac {3 a^{2} c^{2} d \,x^{2}}{2 b^{3}}-\frac {a \,c^{3} x^{2}}{2 b^{2}}-\frac {a^{5} d^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{6}}+\frac {3 a^{4} c \,d^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{5}}-\frac {3 a^{3} c^{2} d \ln \left (b \,x^{2}+a \right )}{2 b^{4}}+\frac {a^{2} c^{3} \ln \left (b \,x^{2}+a \right )}{2 b^{3}} \]

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*x^2*a^4*d^3-3/2/b^4*x^2*a^3*c*
d^2+3/2/b^3*x^2*a^2*c^2*d-1/2/b^2*x^2*a*c^3-1/2*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 = 1.20, size = 219, normalized size = 1.59 \[ \frac {12 \, b^{4} d^{3} x^{10} + 15 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{8} + 20 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{6} + 30 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} - 60 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{2}}{120 \, b^{5}} + \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{6}} \]

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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mupad [B]  time = 0.12, size = 236, normalized size = 1.71 \[ x^4\,\left (\frac {c^3}{4\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{4\,b}\right )-x^8\,\left (\frac {a\,d^3}{8\,b^2}-\frac {3\,c\,d^2}{8\,b}\right )+x^6\,\left (\frac {c^2\,d}{2\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{6\,b}\right )-\frac {\ln \left (b\,x^2+a\right )\,\left (a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3\right )}{2\,b^6}+\frac {d^3\,x^{10}}{10\,b}-\frac {a\,x^2\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.60, size = 201, normalized size = 1.46 \[ - \frac {a^{2} \left (a d - b c\right )^{3} \log {\left (a + b x^{2} \right )}}{2 b^{6}} + x^{8} \left (- \frac {a d^{3}}{8 b^{2}} + \frac {3 c d^{2}}{8 b}\right ) + x^{6} \left (\frac {a^{2} d^{3}}{6 b^{3}} - \frac {a c d^{2}}{2 b^{2}} + \frac {c^{2} d}{2 b}\right ) + x^{4} \left (- \frac {a^{3} d^{3}}{4 b^{4}} + \frac {3 a^{2} c d^{2}}{4 b^{3}} - \frac {3 a c^{2} d}{4 b^{2}} + \frac {c^{3}}{4 b}\right ) + x^{2} \left (\frac {a^{4} d^{3}}{2 b^{5}} - \frac {3 a^{3} c d^{2}}{2 b^{4}} + \frac {3 a^{2} c^{2} d}{2 b^{3}} - \frac {a c^{3}}{2 b^{2}}\right ) + \frac {d^{3} x^{10}}{10 b} \]

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

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