3.1.29 \(\int \frac {(c+d x^2)^3}{(a+b x^2)^2} \, dx\) [29]

Optimal. Leaf size=106 \[ \frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^3}{3 b^2}+\frac {(b c-a d)^3 x}{2 a b^3 \left (a+b x^2\right )}+\frac {(b c-a d)^2 (b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 106, 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 ) (5 a d+b c) (b c-a d)^2}{2 a^{3/2} b^{7/2}}+\frac {d^2 x (3 b c-2 a d)}{b^3}+\frac {x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac {d^3 x^3}{3 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(a + b*x^2)^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.04, size = 106, normalized size = 1.00 \begin {gather*} \frac {d^2 (3 b c-2 a d) x}{b^3}+\frac {d^3 x^3}{3 b^2}+\frac {(b c-a d)^3 x}{2 a b^3 \left (a+b x^2\right )}+\frac {(b c-a d)^2 (b c+5 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^3/(a + b*x^2)^2,x]

[Out]

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

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Maple [A]
time = 0.09, size = 139, normalized size = 1.31

method	result	size
default	\(-\frac {d^{2} \left (-\frac {1}{3} b d \,x^{3}+2 a d x -3 b c x \right )}{b^{3}}+\frac {-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{3}}\)	\(139\)
risch	\(\frac {d^{3} x^{3}}{3 b^{2}}-\frac {2 d^{3} a x}{b^{3}}+\frac {3 d^{2} c x}{b^{2}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{2 a \,b^{3} \left (b \,x^{2}+a \right )}-\frac {5 a^{2} \ln \left (b x +\sqrt {-a b}\right ) d^{3}}{4 b^{3} \sqrt {-a b}}+\frac {9 a \ln \left (b x +\sqrt {-a b}\right ) c \,d^{2}}{4 b^{2} \sqrt {-a b}}-\frac {3 \ln \left (b x +\sqrt {-a b}\right ) c^{2} d}{4 b \sqrt {-a b}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c^{3}}{4 \sqrt {-a b}\, a}+\frac {5 a^{2} \ln \left (-b x +\sqrt {-a b}\right ) d^{3}}{4 b^{3} \sqrt {-a b}}-\frac {9 a \ln \left (-b x +\sqrt {-a b}\right ) c \,d^{2}}{4 b^{2} \sqrt {-a b}}+\frac {3 \ln \left (-b x +\sqrt {-a b}\right ) c^{2} d}{4 b \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c^{3}}{4 \sqrt {-a b}\, a}\)	\(303\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 147, normalized size = 1.39 \begin {gather*} \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {b d^{3} x^{3} + 3 \, {\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x}{3 \, b^{3}} + \frac {{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (92) = 184\).
time = 0.81, size = 442, normalized size = 4.17 \begin {gather*} \left [\frac {4 \, a^{2} b^{3} d^{3} x^{5} + 4 \, {\left (9 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{3} - 3 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 9 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{12 \, {\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}, \frac {2 \, a^{2} b^{3} d^{3} x^{5} + 2 \, {\left (9 \, a^{2} b^{3} c d^{2} - 5 \, a^{3} b^{2} d^{3}\right )} x^{3} + 3 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 9 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x}{6 \, {\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/12*(4*a^2*b^3*d^3*x^5 + 4*(9*a^2*b^3*c*d^2 - 5*a^3*b^2*d^3)*x^3 - 3*(a*b^3*c^3 + 3*a^2*b^2*c^2*d - 9*a^3*b*
c*d^2 + 5*a^4*d^3 + (b^4*c^3 + 3*a*b^3*c^2*d - 9*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*s
qrt(-a*b)*x - a)/(b*x^2 + a)) + 6*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x)/(a^2*b^5*x^
2 + a^3*b^4), 1/6*(2*a^2*b^3*d^3*x^5 + 2*(9*a^2*b^3*c*d^2 - 5*a^3*b^2*d^3)*x^3 + 3*(a*b^3*c^3 + 3*a^2*b^2*c^2*
d - 9*a^3*b*c*d^2 + 5*a^4*d^3 + (b^4*c^3 + 3*a*b^3*c^2*d - 9*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^2)*sqrt(a*b)*arcta
n(sqrt(a*b)*x/a) + 3*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 9*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x)/(a^2*b^5*x^2 + a^3*b^4)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (95) = 190\).
time = 0.59, size = 314, normalized size = 2.96 \begin {gather*} x \left (- \frac {2 a d^{3}}{b^{3}} + \frac {3 c d^{2}}{b^{2}}\right ) + \frac {x \left (- a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}\right )}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \cdot \left (5 a d + b c\right ) \log {\left (- \frac {a^{2} b^{3} \sqrt {- \frac {1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \cdot \left (5 a d + b c\right )}{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \cdot \left (5 a d + b c\right ) \log {\left (\frac {a^{2} b^{3} \sqrt {- \frac {1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \cdot \left (5 a d + b c\right )}{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3}} + x \right )}}{4} + \frac {d^{3} x^{3}}{3 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

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

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Giac [A]
time = 1.12, size = 152, normalized size = 1.43 \begin {gather*} \frac {{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{3}} + \frac {b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \, {\left (b x^{2} + a\right )} a b^{3}} + \frac {b^{4} d^{3} x^{3} + 9 \, b^{4} c d^{2} x - 6 \, a b^{3} d^{3} x}{3 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.10, size = 182, normalized size = 1.72 \begin {gather*} \frac {d^3\,x^3}{3\,b^2}-x\,\left (\frac {2\,a\,d^3}{b^3}-\frac {3\,c\,d^2}{b^2}\right )-\frac {x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,a\,\left (b^4\,x^2+a\,b^3\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+b\,c\right )}{\sqrt {a}\,\left (5\,a^3\,d^3-9\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (5\,a\,d+b\,c\right )}{2\,a^{3/2}\,b^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^3/(a + b*x^2)^2,x)

[Out]

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

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