3.3.9 \(\int \frac {x^4 (c+d x^2)^2}{a+b x^2} \, dx\) [209]

Optimal. Leaf size=105 \[ -\frac {a (b c-a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^3}{3 b^3}+\frac {d (2 b c-a d) x^5}{5 b^2}+\frac {d^2 x^7}{7 b}+\frac {a^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 105, 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 = {472, 211} \begin {gather*} \frac {a^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^2}{b^{9/2}}-\frac {a x (b c-a d)^2}{b^4}+\frac {x^3 (b c-a d)^2}{3 b^3}+\frac {d x^5 (2 b c-a d)}{5 b^2}+\frac {d^2 x^7}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 211

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

Rule 472

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

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Mathematica [A]
time = 0.07, size = 105, normalized size = 1.00 \begin {gather*} -\frac {a (-b c+a d)^2 x}{b^4}+\frac {(b c-a d)^2 x^3}{3 b^3}+\frac {d (2 b c-a d) x^5}{5 b^2}+\frac {d^2 x^7}{7 b}+\frac {a^{3/2} (-b c+a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.13, size = 142, normalized size = 1.35

method	result	size
default	\(-\frac {-\frac {d^{2} x^{7} b^{3}}{7}+\frac {\left (\left (a d -b c \right ) b^{2} d -b^{3} d c \right ) x^{5}}{5}+\frac {\left (\left (a d -b c \right ) b^{2} c -b d \left (a^{2} d -a b c \right )\right ) x^{3}}{3}+\left (a d -b c \right ) \left (a^{2} d -a b c \right ) x}{b^{4}}+\frac {a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{4} \sqrt {a b}}\)	\(142\)
risch	\(\frac {d^{2} x^{7}}{7 b}-\frac {x^{5} a \,d^{2}}{5 b^{2}}+\frac {2 x^{5} d c}{5 b}-\frac {2 x^{3} a c d}{3 b^{2}}+\frac {x^{3} c^{2}}{3 b}+\frac {x^{3} a^{2} d^{2}}{3 b^{3}}-\frac {a^{3} d^{2} x}{b^{4}}+\frac {2 a^{2} c d x}{b^{3}}-\frac {a \,c^{2} x}{b^{2}}+\frac {\sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x +a \right ) d^{2}}{2 b^{5}}-\frac {\sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) c d}{b^{4}}+\frac {\sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x +a \right ) c^{2}}{2 b^{3}}-\frac {\sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x +a \right ) d^{2}}{2 b^{5}}+\frac {\sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x +a \right ) c d}{b^{4}}-\frac {\sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x +a \right ) c^{2}}{2 b^{3}}\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 140, normalized size = 1.33 \begin {gather*} \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{3} d^{2} x^{7} + 21 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{5} + 35 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} - 105 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{105 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.04, size = 304, normalized size = 2.90 \begin {gather*} \left [\frac {30 \, b^{3} d^{2} x^{7} + 42 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{5} + 70 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 105 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 210 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{210 \, b^{4}}, \frac {15 \, b^{3} d^{2} x^{7} + 21 \, {\left (2 \, b^{3} c d - a b^{2} d^{2}\right )} x^{5} + 35 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + 105 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 105 \, {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x}{105 \, b^{4}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (92) = 184\).
time = 0.36, size = 246, normalized size = 2.34 \begin {gather*} x^{5} \left (- \frac {a d^{2}}{5 b^{2}} + \frac {2 c d}{5 b}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{3 b^{3}} - \frac {2 a c d}{3 b^{2}} + \frac {c^{2}}{3 b}\right ) + x \left (- \frac {a^{3} d^{2}}{b^{4}} + \frac {2 a^{2} c d}{b^{3}} - \frac {a c^{2}}{b^{2}}\right ) - \frac {\sqrt {- \frac {a^{3}}{b^{9}}} \left (a d - b c\right )^{2} \log {\left (- \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}} \left (a d - b c\right )^{2}}{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {a^{3}}{b^{9}}} \left (a d - b c\right )^{2} \log {\left (\frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}} \left (a d - b c\right )^{2}}{a^{3} d^{2} - 2 a^{2} b c d + a b^{2} c^{2}} + x \right )}}{2} + \frac {d^{2} x^{7}}{7 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.95, size = 153, normalized size = 1.46 \begin {gather*} \frac {{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{6} d^{2} x^{7} + 42 \, b^{6} c d x^{5} - 21 \, a b^{5} d^{2} x^{5} + 35 \, b^{6} c^{2} x^{3} - 70 \, a b^{5} c d x^{3} + 35 \, a^{2} b^{4} d^{2} x^{3} - 105 \, a b^{5} c^{2} x + 210 \, a^{2} b^{4} c d x - 105 \, a^{3} b^{3} d^{2} x}{105 \, b^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.06, size = 169, normalized size = 1.61 \begin {gather*} x^3\,\left (\frac {c^2}{3\,b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{3\,b}\right )-x^5\,\left (\frac {a\,d^2}{5\,b^2}-\frac {2\,c\,d}{5\,b}\right )+\frac {d^2\,x^7}{7\,b}+\frac {a^{3/2}\,\mathrm {atan}\left (\frac {a^{3/2}\,\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2}{a^4\,d^2-2\,a^3\,b\,c\,d+a^2\,b^2\,c^2}\right )\,{\left (a\,d-b\,c\right )}^2}{b^{9/2}}-\frac {a\,x\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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