3.223 \(\int \frac {(c+d x^2)^3}{a+b x^2} \, dx\)

Optimal. Leaf size=98 \[ \frac {d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}+\frac {d^2 x^3 (3 b c-a d)}{3 b^2}+\frac {d^3 x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {390, 205} \[ \frac {d x \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {d^2 x^3 (3 b c-a d)}{3 b^2}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}}+\frac {d^3 x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(a + b*x^2),x]

[Out]

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

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 390

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

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Mathematica [A]  time = 0.07, size = 92, normalized size = 0.94 \[ \frac {d x \left (15 a^2 d^2-5 a b d \left (9 c+d x^2\right )+3 b^2 \left (15 c^2+5 c d x^2+d^2 x^4\right )\right )}{15 b^3}+\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*(15*a^2*d^2 - 5*a*b*d*(9*c + d*x^2) + 3*b^2*(15*c^2 + 5*c*d*x^2 + d^2*x^4)))/(15*b^3) + ((b*c - a*d)^3*Ar
cTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*b^(7/2))

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fricas [A]  time = 0.50, size = 292, normalized size = 2.98 \[ \left [\frac {6 \, a b^{3} d^{3} x^{5} + 10 \, {\left (3 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 30 \, {\left (3 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x}{30 \, a b^{4}}, \frac {3 \, a b^{3} d^{3} x^{5} + 5 \, {\left (3 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{3} + 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (3 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x}{15 \, a b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.36, size = 129, normalized size = 1.32 \[ \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{4} d^{3} x^{5} + 15 \, b^{4} c d^{2} x^{3} - 5 \, a b^{3} d^{3} x^{3} + 45 \, b^{4} c^{2} d x - 45 \, a b^{3} c d^{2} x + 15 \, a^{2} b^{2} d^{3} x}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*b^4*d^3*x^
5 + 15*b^4*c*d^2*x^3 - 5*a*b^3*d^3*x^3 + 45*b^4*c^2*d*x - 45*a*b^3*c*d^2*x + 15*a^2*b^2*d^3*x)/b^5

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maple [A]  time = 0.00, size = 161, normalized size = 1.64 \[ \frac {d^{3} x^{5}}{5 b}-\frac {a \,d^{3} x^{3}}{3 b^{2}}+\frac {c \,d^{2} x^{3}}{b}-\frac {a^{3} d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {3 a^{2} c \,d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {3 a \,c^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {c^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}+\frac {a^{2} d^{3} x}{b^{3}}-\frac {3 a c \,d^{2} x}{b^{2}}+\frac {3 c^{2} d x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.43, size = 122, normalized size = 1.24 \[ \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{2} d^{3} x^{5} + 5 \, {\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{3} + 15 \, {\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x}{15 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/15*(3*b^2*d^3*x^
5 + 5*(3*b^2*c*d^2 - a*b*d^3)*x^3 + 15*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*x)/b^3

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mupad [B]  time = 0.12, size = 146, normalized size = 1.49 \[ x\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )-x^3\,\left (\frac {a\,d^3}{3\,b^2}-\frac {c\,d^2}{b}\right )+\frac {d^3\,x^5}{5\,b}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{\sqrt {a}\,b^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^3/(a + b*x^2),x)

[Out]

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

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sympy [B]  time = 0.60, size = 238, normalized size = 2.43 \[ x^{3} \left (- \frac {a d^{3}}{3 b^{2}} + \frac {c d^{2}}{b}\right ) + x \left (\frac {a^{2} d^{3}}{b^{3}} - \frac {3 a c d^{2}}{b^{2}} + \frac {3 c^{2} d}{b}\right ) + \frac {\sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right )^{3} \log {\left (- \frac {a b^{3} \sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right )^{3} \log {\left (\frac {a b^{3} \sqrt {- \frac {1}{a b^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {d^{3} x^{5}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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