3.1.28 \(\int \frac {(c+d x^2)^4}{(a+b x^2)^2} \, dx\) [28]

Optimal. Leaf size=142 \[ \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac {d^4 x^5}{5 b^2}+\frac {(b c-a d)^4 x}{2 a b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^3 (b c+7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{9/2}} \]

[Out]

d^2*(3*a^2*d^2-8*a*b*c*d+6*b^2*c^2)*x/b^4+2/3*d^3*(-a*d+2*b*c)*x^3/b^3+1/5*d^4*x^5/b^2+1/2*(-a*d+b*c)^4*x/a/b^
4/(b*x^2+a)+1/2*(-a*d+b*c)^3*(7*a*d+b*c)*arctan(x*b^(1/2)/a^(1/2))/a^(3/2)/b^(9/2)

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Rubi [A]
time = 0.08, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {398, 393, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^3 (7 a d+b c)}{2 a^{3/2} b^{9/2}}+\frac {d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac {x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac {2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac {d^4 x^5}{5 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4/(a + b*x^2)^2,x]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b^3) + (d^4*x^5)/(5*b^2) + ((b*
c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*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 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^2} \, dx &=\int \left (\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right )}{b^4}+\frac {2 d^3 (2 b c-a d) x^2}{b^3}+\frac {d^4 x^4}{b^2}+\frac {(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^2}{b^4 \left (a+b x^2\right )^2}\right ) \, dx\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac {d^4 x^5}{5 b^2}+\frac {\int \frac {(b c-a d)^3 (b c+3 a d)+4 b d (b c-a d)^3 x^2}{\left (a+b x^2\right )^2} \, dx}{b^4}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac {d^4 x^5}{5 b^2}+\frac {(b c-a d)^4 x}{2 a b^4 \left (a+b x^2\right )}+\frac {\left ((b c-a d)^3 (b c+7 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a b^4}\\ &=\frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac {d^4 x^5}{5 b^2}+\frac {(b c-a d)^4 x}{2 a b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^3 (b c+7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 142, normalized size = 1.00 \begin {gather*} \frac {d^2 \left (6 b^2 c^2-8 a b c d+3 a^2 d^2\right ) x}{b^4}+\frac {2 d^3 (2 b c-a d) x^3}{3 b^3}+\frac {d^4 x^5}{5 b^2}+\frac {(b c-a d)^4 x}{2 a b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^3 (b c+7 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b^3) + (d^4*x^5)/(5*b^2) + ((b*
c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(
9/2))

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Maple [A]
time = 0.11, size = 206, normalized size = 1.45

method	result	size
default	\(\frac {d^{2} \left (\frac {1}{5} b^{2} x^{5} d^{2}-\frac {2}{3} a b \,d^{2} x^{3}+\frac {4}{3} b^{2} c d \,x^{3}+3 a^{2} d^{2} x -8 a b c d x +6 b^{2} c^{2} x \right )}{b^{4}}-\frac {-\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (7 a^{4} d^{4}-20 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{4}}\)	\(206\)
risch	\(\frac {d^{4} x^{5}}{5 b^{2}}-\frac {2 d^{4} a \,x^{3}}{3 b^{3}}+\frac {4 d^{3} c \,x^{3}}{3 b^{2}}+\frac {3 d^{4} a^{2} x}{b^{4}}-\frac {8 d^{3} a c x}{b^{3}}+\frac {6 d^{2} c^{2} x}{b^{2}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x}{2 a \,b^{4} \left (b \,x^{2}+a \right )}-\frac {7 a^{3} \ln \left (b x -\sqrt {-a b}\right ) d^{4}}{4 b^{4} \sqrt {-a b}}+\frac {5 a^{2} \ln \left (b x -\sqrt {-a b}\right ) c \,d^{3}}{b^{3} \sqrt {-a b}}-\frac {9 a \ln \left (b x -\sqrt {-a b}\right ) c^{2} d^{2}}{2 b^{2} \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c^{3} d}{b \sqrt {-a b}}+\frac {\ln \left (b x -\sqrt {-a b}\right ) c^{4}}{4 \sqrt {-a b}\, a}+\frac {7 a^{3} \ln \left (-b x -\sqrt {-a b}\right ) d^{4}}{4 b^{4} \sqrt {-a b}}-\frac {5 a^{2} \ln \left (-b x -\sqrt {-a b}\right ) c \,d^{3}}{b^{3} \sqrt {-a b}}+\frac {9 a \ln \left (-b x -\sqrt {-a b}\right ) c^{2} d^{2}}{2 b^{2} \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c^{3} d}{b \sqrt {-a b}}-\frac {\ln \left (-b x -\sqrt {-a b}\right ) c^{4}}{4 \sqrt {-a b}\, a}\)	\(437\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d^2/b^4*(1/5*b^2*x^5*d^2-2/3*a*b*d^2*x^3+4/3*b^2*c*d*x^3+3*a^2*d^2*x-8*a*b*c*d*x+6*b^2*c^2*x)-1/b^4*(-1/2*(a^4
*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)/a*x/(b*x^2+a)+1/2*(7*a^4*d^4-20*a^3*b*c*d^3+18*a^2
*b^2*c^2*d^2-4*a*b^3*c^3*d-b^4*c^4)/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))

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Maxima [A]
time = 0.50, size = 213, normalized size = 1.50 \begin {gather*} \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} x}{2 \, {\left (a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac {3 \, b^{2} d^{4} x^{5} + 10 \, {\left (2 \, b^{2} c d^{3} - a b d^{4}\right )} x^{3} + 15 \, {\left (6 \, b^{2} c^{2} d^{2} - 8 \, a b c d^{3} + 3 \, a^{2} d^{4}\right )} x}{15 \, b^{4}} + \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*x/(a*b^5*x^2 + a^2*b^4) + 1/15*(3*
b^2*d^4*x^5 + 10*(2*b^2*c*d^3 - a*b*d^4)*x^3 + 15*(6*b^2*c^2*d^2 - 8*a*b*c*d^3 + 3*a^2*d^4)*x)/b^4 + 1/2*(b^4*
c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (126) = 252\).
time = 0.65, size = 612, normalized size = 4.31 \begin {gather*} \left [\frac {12 \, a^{2} b^{4} d^{4} x^{7} + 4 \, {\left (20 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 20 \, {\left (18 \, a^{2} b^{4} c^{2} d^{2} - 20 \, a^{3} b^{3} c d^{3} + 7 \, a^{4} b^{2} d^{4}\right )} x^{3} + 15 \, {\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 20 \, a^{4} b c d^{3} - 7 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 18 \, a^{2} b^{3} c^{2} d^{2} + 20 \, a^{3} b^{2} c d^{3} - 7 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 30 \, {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 20 \, a^{4} b^{2} c d^{3} + 7 \, a^{5} b d^{4}\right )} x}{60 \, {\left (a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}, \frac {6 \, a^{2} b^{4} d^{4} x^{7} + 2 \, {\left (20 \, a^{2} b^{4} c d^{3} - 7 \, a^{3} b^{3} d^{4}\right )} x^{5} + 10 \, {\left (18 \, a^{2} b^{4} c^{2} d^{2} - 20 \, a^{3} b^{3} c d^{3} + 7 \, a^{4} b^{2} d^{4}\right )} x^{3} + 15 \, {\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 20 \, a^{4} b c d^{3} - 7 \, a^{5} d^{4} + {\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 18 \, a^{2} b^{3} c^{2} d^{2} + 20 \, a^{3} b^{2} c d^{3} - 7 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 20 \, a^{4} b^{2} c d^{3} + 7 \, a^{5} b d^{4}\right )} x}{30 \, {\left (a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/60*(12*a^2*b^4*d^4*x^7 + 4*(20*a^2*b^4*c*d^3 - 7*a^3*b^3*d^4)*x^5 + 20*(18*a^2*b^4*c^2*d^2 - 20*a^3*b^3*c*d
^3 + 7*a^4*b^2*d^4)*x^3 + 15*(a*b^4*c^4 + 4*a^2*b^3*c^3*d - 18*a^3*b^2*c^2*d^2 + 20*a^4*b*c*d^3 - 7*a^5*d^4 +
(b^5*c^4 + 4*a*b^4*c^3*d - 18*a^2*b^3*c^2*d^2 + 20*a^3*b^2*c*d^3 - 7*a^4*b*d^4)*x^2)*sqrt(-a*b)*log((b*x^2 + 2
*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 30*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 18*a^3*b^3*c^2*d^2 - 20*a^4*b^2*c*d^3 + 7*
a^5*b*d^4)*x)/(a^2*b^6*x^2 + a^3*b^5), 1/30*(6*a^2*b^4*d^4*x^7 + 2*(20*a^2*b^4*c*d^3 - 7*a^3*b^3*d^4)*x^5 + 10
*(18*a^2*b^4*c^2*d^2 - 20*a^3*b^3*c*d^3 + 7*a^4*b^2*d^4)*x^3 + 15*(a*b^4*c^4 + 4*a^2*b^3*c^3*d - 18*a^3*b^2*c^
2*d^2 + 20*a^4*b*c*d^3 - 7*a^5*d^4 + (b^5*c^4 + 4*a*b^4*c^3*d - 18*a^2*b^3*c^2*d^2 + 20*a^3*b^2*c*d^3 - 7*a^4*
b*d^4)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + 15*(a*b^5*c^4 - 4*a^2*b^4*c^3*d + 18*a^3*b^3*c^2*d^2 - 20*a^4*b^
2*c*d^3 + 7*a^5*b*d^4)*x)/(a^2*b^6*x^2 + a^3*b^5)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (133) = 266\).
time = 0.82, size = 403, normalized size = 2.84 \begin {gather*} x^{3} \left (- \frac {2 a d^{4}}{3 b^{3}} + \frac {4 c d^{3}}{3 b^{2}}\right ) + x \left (\frac {3 a^{2} d^{4}}{b^{4}} - \frac {8 a c d^{3}}{b^{3}} + \frac {6 c^{2} d^{2}}{b^{2}}\right ) + \frac {x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{2 a^{2} b^{4} + 2 a b^{5} x^{2}} + \frac {\sqrt {- \frac {1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \cdot \left (7 a d + b c\right ) \log {\left (- \frac {a^{2} b^{4} \sqrt {- \frac {1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \cdot \left (7 a d + b c\right )}{7 a^{4} d^{4} - 20 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \cdot \left (7 a d + b c\right ) \log {\left (\frac {a^{2} b^{4} \sqrt {- \frac {1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \cdot \left (7 a d + b c\right )}{7 a^{4} d^{4} - 20 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4}} + x \right )}}{4} + \frac {d^{4} x^{5}}{5 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*(-2*a*d**4/(3*b**3) + 4*c*d**3/(3*b**2)) + x*(3*a**2*d**4/b**4 - 8*a*c*d**3/b**3 + 6*c**2*d**2/b**2) + x*
(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*c**4)/(2*a**2*b**4 + 2*a*b**5*x*
*2) + sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b*c)*log(-a**2*b**4*sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*
a*d + b*c)/(7*a**4*d**4 - 20*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d - b**4*c**4) + x)/4 - sq
rt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b*c)*log(a**2*b**4*sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b*c
)/(7*a**4*d**4 - 20*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d - b**4*c**4) + x)/4 + d**4*x**5/(
5*b**2)

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Giac [A]
time = 2.22, size = 220, normalized size = 1.55 \begin {gather*} \frac {{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{4}} + \frac {b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{2 \, {\left (b x^{2} + a\right )} a b^{4}} + \frac {3 \, b^{8} d^{4} x^{5} + 20 \, b^{8} c d^{3} x^{3} - 10 \, a b^{7} d^{4} x^{3} + 90 \, b^{8} c^{2} d^{2} x - 120 \, a b^{7} c d^{3} x + 45 \, a^{2} b^{6} d^{4} x}{15 \, b^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)*arctan(b*x/sqrt(a*b))/(sqrt(a*
b)*a*b^4) + 1/2*(b^4*c^4*x - 4*a*b^3*c^3*d*x + 6*a^2*b^2*c^2*d^2*x - 4*a^3*b*c*d^3*x + a^4*d^4*x)/((b*x^2 + a)
*a*b^4) + 1/15*(3*b^8*d^4*x^5 + 20*b^8*c*d^3*x^3 - 10*a*b^7*d^4*x^3 + 90*b^8*c^2*d^2*x - 120*a*b^7*c*d^3*x + 4
5*a^2*b^6*d^4*x)/b^10

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Mupad [B]
time = 5.05, size = 261, normalized size = 1.84 \begin {gather*} x\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^4}{b^3}-\frac {4\,c\,d^3}{b^2}\right )}{b}-\frac {a^2\,d^4}{b^4}+\frac {6\,c^2\,d^2}{b^2}\right )-x^3\,\left (\frac {2\,a\,d^4}{3\,b^3}-\frac {4\,c\,d^3}{3\,b^2}\right )+\frac {d^4\,x^5}{5\,b^2}+\frac {x\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{2\,a\,\left (b^5\,x^2+a\,b^4\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3\,\left (7\,a\,d+b\,c\right )}{\sqrt {a}\,\left (-7\,a^4\,d^4+20\,a^3\,b\,c\,d^3-18\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (7\,a\,d+b\,c\right )}{2\,a^{3/2}\,b^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^4/(a + b*x^2)^2,x)

[Out]

x*((2*a*((2*a*d^4)/b^3 - (4*c*d^3)/b^2))/b - (a^2*d^4)/b^4 + (6*c^2*d^2)/b^2) - x^3*((2*a*d^4)/(3*b^3) - (4*c*
d^3)/(3*b^2)) + (d^4*x^5)/(5*b^2) + (x*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)
)/(2*a*(a*b^4 + b^5*x^2)) + (atan((b^(1/2)*x*(a*d - b*c)^3*(7*a*d + b*c))/(a^(1/2)*(b^4*c^4 - 7*a^4*d^4 - 18*a
^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 20*a^3*b*c*d^3)))*(a*d - b*c)^3*(7*a*d + b*c))/(2*a^(3/2)*b^(9/2))

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