3.3.31 \(\int \frac {x^2}{(a+b x^2) (c+d x^2)} \, dx\) [231]

Optimal. Leaf size=70 \[ -\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)} \]

[Out]

-arctan(x*b^(1/2)/a^(1/2))*a^(1/2)/(-a*d+b*c)/b^(1/2)+arctan(x*d^(1/2)/c^(1/2))*c^(1/2)/(-a*d+b*c)/d^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 70, 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 = {492, 211} \begin {gather*} \frac {\sqrt {c} \text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d} (b c-a d)}-\frac {\sqrt {a} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/((a + b*x^2)*(c + d*x^2)),x]

[Out]

-((Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*(b*c - a*d))) + (Sqrt[c]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[d
]*(b*c - a*d))

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 492

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

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 61, normalized size = 0.87 \begin {gather*} \frac {-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {d}}}{b c-a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b]) + (Sqrt[c]*ArcTan[(Sqrt[d]*x)/Sqrt[c]])/Sqrt[d])/(b*c - a*d)

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Maple [A]
time = 0.13, size = 55, normalized size = 0.79

method	result	size
default	\(\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\left (a d -b c \right ) \sqrt {a b}}-\frac {c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\left (a d -b c \right ) \sqrt {c d}}\)	\(55\)
risch	\(\frac {\sqrt {-a b}\, \ln \left (\left (-\left (-a b \right )^{\frac {3}{2}} a \,d^{2}-\left (-a b \right )^{\frac {3}{2}} b c d -a^{2} \sqrt {-a b}\, d^{2} b -b^{3} c^{2} \sqrt {-a b}\right ) x -a^{2} b^{2} c d +a \,b^{3} c^{2}\right )}{2 b \left (a d -b c \right )}-\frac {\sqrt {-a b}\, \ln \left (\left (\left (-a b \right )^{\frac {3}{2}} a \,d^{2}+\left (-a b \right )^{\frac {3}{2}} b c d +a^{2} \sqrt {-a b}\, d^{2} b +b^{3} c^{2} \sqrt {-a b}\right ) x -a^{2} b^{2} c d +a \,b^{3} c^{2}\right )}{2 b \left (a d -b c \right )}+\frac {\sqrt {-c d}\, \ln \left (\left (-\left (-c d \right )^{\frac {3}{2}} a b d -\left (-c d \right )^{\frac {3}{2}} b^{2} c -a^{2} \sqrt {-c d}\, d^{3}-b^{2} c^{2} \sqrt {-c d}\, d \right ) x -a^{2} c \,d^{3}+a b \,c^{2} d^{2}\right )}{2 d \left (a d -b c \right )}-\frac {\sqrt {-c d}\, \ln \left (\left (\left (-c d \right )^{\frac {3}{2}} a b d +\left (-c d \right )^{\frac {3}{2}} b^{2} c +a^{2} \sqrt {-c d}\, d^{3}+b^{2} c^{2} \sqrt {-c d}\, d \right ) x -a^{2} c \,d^{3}+a b \,c^{2} d^{2}\right )}{2 d \left (a d -b c \right )}\)	\(378\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a/(a*d-b*c)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))-c/(a*d-b*c)/(c*d)^(1/2)*arctan(d*x/(c*d)^(1/2))

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Maxima [A]
time = 0.52, size = 54, normalized size = 0.77 \begin {gather*} -\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c - a d\right )} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) + c*arctan(d*x/sqrt(c*d))/((b*c - a*d)*sqrt(c*d))

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Fricas [A]
time = 1.23, size = 309, normalized size = 4.41 \begin {gather*} \left [-\frac {\sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} - 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right )}{2 \, {\left (b c - a d\right )}}, -\frac {2 \, \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + \sqrt {-\frac {c}{d}} \log \left (\frac {d x^{2} - 2 \, d x \sqrt {-\frac {c}{d}} - c}{d x^{2} + c}\right )}{2 \, {\left (b c - a d\right )}}, \frac {2 \, \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right ) - \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{2 \, {\left (b c - a d\right )}}, -\frac {\sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - \sqrt {\frac {c}{d}} \arctan \left (\frac {d x \sqrt {\frac {c}{d}}}{c}\right )}{b c - a d}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) + sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(-c/d) -
c)/(d*x^2 + c)))/(b*c - a*d), -1/2*(2*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) + sqrt(-c/d)*log((d*x^2 - 2*d*x*sqrt(
-c/d) - c)/(d*x^2 + c)))/(b*c - a*d), 1/2*(2*sqrt(c/d)*arctan(d*x*sqrt(c/d)/c) - sqrt(-a/b)*log((b*x^2 + 2*b*x
*sqrt(-a/b) - a)/(b*x^2 + a)))/(b*c - a*d), -(sqrt(a/b)*arctan(b*x*sqrt(a/b)/a) - sqrt(c/d)*arctan(d*x*sqrt(c/
d)/c))/(b*c - a*d)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (60) = 120\).
time = 1.83, size = 570, normalized size = 8.14 \begin {gather*} \frac {\sqrt {- \frac {a}{b}} \log {\left (- \frac {2 a^{2} b d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a d \sqrt {- \frac {a}{b}}}{a d - b c} - \frac {2 b^{3} c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b c \sqrt {- \frac {a}{b}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {a}{b}} \log {\left (\frac {2 a^{2} b d^{3} \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{2} c d^{2} \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a d \sqrt {- \frac {a}{b}}}{a d - b c} + \frac {2 b^{3} c^{2} d \left (- \frac {a}{b}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b c \sqrt {- \frac {a}{b}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} + \frac {\sqrt {- \frac {c}{d}} \log {\left (- \frac {2 a^{2} b d^{3} \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {4 a b^{2} c d^{2} \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {a d \sqrt {- \frac {c}{d}}}{a d - b c} - \frac {2 b^{3} c^{2} d \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {b c \sqrt {- \frac {c}{d}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} - \frac {\sqrt {- \frac {c}{d}} \log {\left (\frac {2 a^{2} b d^{3} \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} - \frac {4 a b^{2} c d^{2} \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {a d \sqrt {- \frac {c}{d}}}{a d - b c} + \frac {2 b^{3} c^{2} d \left (- \frac {c}{d}\right )^{\frac {3}{2}}}{\left (a d - b c\right )^{3}} + \frac {b c \sqrt {- \frac {c}{d}}}{a d - b c} + x \right )}}{2 \left (a d - b c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.59, size = 54, normalized size = 0.77 \begin {gather*} -\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} {\left (b c - a d\right )}} + \frac {c \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{{\left (b c - a d\right )} \sqrt {c d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*(b*c - a*d)) + c*arctan(d*x/sqrt(c*d))/((b*c - a*d)*sqrt(c*d))

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Mupad [B]
time = 0.19, size = 133, normalized size = 1.90 \begin {gather*} \frac {\ln \left (a+x\,\sqrt {-a\,b}\right )\,\sqrt {-a\,b}}{2\,b^2\,c-2\,a\,b\,d}-\frac {\ln \left (a-x\,\sqrt {-a\,b}\right )\,\sqrt {-a\,b}}{2\,\left (b^2\,c-a\,b\,d\right )}-\frac {\ln \left (c-x\,\sqrt {-c\,d}\right )\,\sqrt {-c\,d}}{2\,\left (a\,d^2-b\,c\,d\right )}+\frac {\ln \left (c+x\,\sqrt {-c\,d}\right )\,\sqrt {-c\,d}}{2\,a\,d^2-2\,b\,c\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/((a + b*x^2)*(c + d*x^2)),x)

[Out]

(log(a + x*(-a*b)^(1/2))*(-a*b)^(1/2))/(2*b^2*c - 2*a*b*d) - (log(a - x*(-a*b)^(1/2))*(-a*b)^(1/2))/(2*(b^2*c
- a*b*d)) - (log(c - x*(-c*d)^(1/2))*(-c*d)^(1/2))/(2*(a*d^2 - b*c*d)) + (log(c + x*(-c*d)^(1/2))*(-c*d)^(1/2)
)/(2*a*d^2 - 2*b*c*d)

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