3.168 \(\int \frac {x^4 (a+b x^2)^2}{c+d x^2} \, dx\)

Optimal. Leaf size=104 \[ \frac {c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{9/2}}-\frac {c x (b c-a d)^2}{d^4}+\frac {x^3 (b c-a d)^2}{3 d^3}-\frac {b x^5 (b c-2 a d)}{5 d^2}+\frac {b^2 x^7}{7 d} \]

[Out]

-c*(-a*d+b*c)^2*x/d^4+1/3*(-a*d+b*c)^2*x^3/d^3-1/5*b*(-2*a*d+b*c)*x^5/d^2+1/7*b^2*x^7/d+c^(3/2)*(-a*d+b*c)^2*a
rctan(x*d^(1/2)/c^(1/2))/d^(9/2)

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Rubi [A]  time = 0.07, antiderivative size = 104, 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 = {461, 205} \[ \frac {c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{9/2}}-\frac {b x^5 (b c-2 a d)}{5 d^2}+\frac {x^3 (b c-a d)^2}{3 d^3}-\frac {c x (b c-a d)^2}{d^4}+\frac {b^2 x^7}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*x^2)^2)/(c + d*x^2),x]

[Out]

-((c*(b*c - a*d)^2*x)/d^4) + ((b*c - a*d)^2*x^3)/(3*d^3) - (b*(b*c - 2*a*d)*x^5)/(5*d^2) + (b^2*x^7)/(7*d) + (
c^(3/2)*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(9/2)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rubi steps

\begin {align*} \int \frac {x^4 \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac {c (b c-a d)^2}{d^4}+\frac {(b c-a d)^2 x^2}{d^3}-\frac {b (b c-2 a d) x^4}{d^2}+\frac {b^2 x^6}{d}+\frac {b^2 c^4-2 a b c^3 d+a^2 c^2 d^2}{d^4 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac {c (b c-a d)^2 x}{d^4}+\frac {(b c-a d)^2 x^3}{3 d^3}-\frac {b (b c-2 a d) x^5}{5 d^2}+\frac {b^2 x^7}{7 d}+\frac {\left (c^2 (b c-a d)^2\right ) \int \frac {1}{c+d x^2} \, dx}{d^4}\\ &=-\frac {c (b c-a d)^2 x}{d^4}+\frac {(b c-a d)^2 x^3}{3 d^3}-\frac {b (b c-2 a d) x^5}{5 d^2}+\frac {b^2 x^7}{7 d}+\frac {c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 104, normalized size = 1.00 \[ \frac {c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{d^{9/2}}-\frac {c x (b c-a d)^2}{d^4}+\frac {x^3 (a d-b c)^2}{3 d^3}-\frac {b x^5 (b c-2 a d)}{5 d^2}+\frac {b^2 x^7}{7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(a + b*x^2)^2)/(c + d*x^2),x]

[Out]

-((c*(b*c - a*d)^2*x)/d^4) + ((-(b*c) + a*d)^2*x^3)/(3*d^3) - (b*(b*c - 2*a*d)*x^5)/(5*d^2) + (b^2*x^7)/(7*d)
+ (c^(3/2)*(b*c - a*d)^2*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/d^(9/2)

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fricas [A]  time = 0.48, size = 302, normalized size = 2.90 \[ \left [\frac {30 \, b^{2} d^{3} x^{7} - 42 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 70 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} + 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right ) - 210 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{210 \, d^{4}}, \frac {15 \, b^{2} d^{3} x^{7} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{105 \, d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^2/(d*x^2+c),x, algorithm="fricas")

[Out]

[1/210*(30*b^2*d^3*x^7 - 42*(b^2*c*d^2 - 2*a*b*d^3)*x^5 + 70*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^3 + 105*(b^
2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(-c/d)*log((d*x^2 + 2*d*x*sqrt(-c/d) - c)/(d*x^2 + c)) - 210*(b^2*c^3 - 2
*a*b*c^2*d + a^2*c*d^2)*x)/d^4, 1/105*(15*b^2*d^3*x^7 - 21*(b^2*c*d^2 - 2*a*b*d^3)*x^5 + 35*(b^2*c^2*d - 2*a*b
*c*d^2 + a^2*d^3)*x^3 + 105*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) - 105*(b^2*c
^3 - 2*a*b*c^2*d + a^2*c*d^2)*x)/d^4]

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giac [A]  time = 0.31, size = 153, normalized size = 1.47 \[ \frac {{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{4}} + \frac {15 \, b^{2} d^{6} x^{7} - 21 \, b^{2} c d^{5} x^{5} + 42 \, a b d^{6} x^{5} + 35 \, b^{2} c^{2} d^{4} x^{3} - 70 \, a b c d^{5} x^{3} + 35 \, a^{2} d^{6} x^{3} - 105 \, b^{2} c^{3} d^{3} x + 210 \, a b c^{2} d^{4} x - 105 \, a^{2} c d^{5} x}{105 \, d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^2/(d*x^2+c),x, algorithm="giac")

[Out]

(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*d^4) + 1/105*(15*b^2*d^6*x^7 - 21*b^2*c
*d^5*x^5 + 42*a*b*d^6*x^5 + 35*b^2*c^2*d^4*x^3 - 70*a*b*c*d^5*x^3 + 35*a^2*d^6*x^3 - 105*b^2*c^3*d^3*x + 210*a
*b*c^2*d^4*x - 105*a^2*c*d^5*x)/d^7

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maple [A]  time = 0.01, size = 176, normalized size = 1.69 \[ \frac {b^{2} x^{7}}{7 d}+\frac {2 a b \,x^{5}}{5 d}-\frac {b^{2} c \,x^{5}}{5 d^{2}}+\frac {a^{2} x^{3}}{3 d}-\frac {2 a b c \,x^{3}}{3 d^{2}}+\frac {b^{2} c^{2} x^{3}}{3 d^{3}}+\frac {a^{2} c^{2} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{2}}-\frac {2 a b \,c^{3} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{3}}+\frac {b^{2} c^{4} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d}\, d^{4}}-\frac {a^{2} c x}{d^{2}}+\frac {2 a b \,c^{2} x}{d^{3}}-\frac {b^{2} c^{3} x}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)^2/(d*x^2+c),x)

[Out]

1/7*b^2*x^7/d+2/5/d*x^5*a*b-1/5/d^2*x^5*b^2*c+1/3/d*x^3*a^2-2/3/d^2*x^3*a*b*c+1/3/d^3*x^3*b^2*c^2-1/d^2*a^2*c*
x+2/d^3*a*b*c^2*x-1/d^4*b^2*c^3*x+c^2/d^2/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2))*a^2-2*c^3/d^3/(c*d)^(1/2)*arctan
(d*x/(c*d)^(1/2))*a*b+c^4/d^4/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2))*b^2

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maxima [A]  time = 2.41, size = 139, normalized size = 1.34 \[ \frac {{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{4}} + \frac {15 \, b^{2} d^{3} x^{7} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{105 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^2/(d*x^2+c),x, algorithm="maxima")

[Out]

(b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*arctan(d*x/sqrt(c*d))/(sqrt(c*d)*d^4) + 1/105*(15*b^2*d^3*x^7 - 21*(b^2*
c*d^2 - 2*a*b*d^3)*x^5 + 35*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x^3 - 105*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*
x)/d^4

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mupad [B]  time = 0.06, size = 169, normalized size = 1.62 \[ x^3\,\left (\frac {a^2}{3\,d}+\frac {c\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )}{3\,d}\right )-x^5\,\left (\frac {b^2\,c}{5\,d^2}-\frac {2\,a\,b}{5\,d}\right )+\frac {b^2\,x^7}{7\,d}+\frac {c^{3/2}\,\mathrm {atan}\left (\frac {c^{3/2}\,\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^2}{a^2\,c^2\,d^2-2\,a\,b\,c^3\,d+b^2\,c^4}\right )\,{\left (a\,d-b\,c\right )}^2}{d^{9/2}}-\frac {c\,x\,\left (\frac {a^2}{d}+\frac {c\,\left (\frac {b^2\,c}{d^2}-\frac {2\,a\,b}{d}\right )}{d}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(a + b*x^2)^2)/(c + d*x^2),x)

[Out]

x^3*(a^2/(3*d) + (c*((b^2*c)/d^2 - (2*a*b)/d))/(3*d)) - x^5*((b^2*c)/(5*d^2) - (2*a*b)/(5*d)) + (b^2*x^7)/(7*d
) + (c^(3/2)*atan((c^(3/2)*d^(1/2)*x*(a*d - b*c)^2)/(b^2*c^4 + a^2*c^2*d^2 - 2*a*b*c^3*d))*(a*d - b*c)^2)/d^(9
/2) - (c*x*(a^2/d + (c*((b^2*c)/d^2 - (2*a*b)/d))/d))/d

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sympy [B]  time = 0.53, size = 246, normalized size = 2.37 \[ \frac {b^{2} x^{7}}{7 d} + x^{5} \left (\frac {2 a b}{5 d} - \frac {b^{2} c}{5 d^{2}}\right ) + x^{3} \left (\frac {a^{2}}{3 d} - \frac {2 a b c}{3 d^{2}} + \frac {b^{2} c^{2}}{3 d^{3}}\right ) + x \left (- \frac {a^{2} c}{d^{2}} + \frac {2 a b c^{2}}{d^{3}} - \frac {b^{2} c^{3}}{d^{4}}\right ) - \frac {\sqrt {- \frac {c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log {\left (- \frac {d^{4} \sqrt {- \frac {c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log {\left (\frac {d^{4} \sqrt {- \frac {c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**2/(d*x**2+c),x)

[Out]

b**2*x**7/(7*d) + x**5*(2*a*b/(5*d) - b**2*c/(5*d**2)) + x**3*(a**2/(3*d) - 2*a*b*c/(3*d**2) + b**2*c**2/(3*d*
*3)) + x*(-a**2*c/d**2 + 2*a*b*c**2/d**3 - b**2*c**3/d**4) - sqrt(-c**3/d**9)*(a*d - b*c)**2*log(-d**4*sqrt(-c
**3/d**9)*(a*d - b*c)**2/(a**2*c*d**2 - 2*a*b*c**2*d + b**2*c**3) + x)/2 + sqrt(-c**3/d**9)*(a*d - b*c)**2*log
(d**4*sqrt(-c**3/d**9)*(a*d - b*c)**2/(a**2*c*d**2 - 2*a*b*c**2*d + b**2*c**3) + x)/2

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