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

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

[Out]

b*(3*a^2*d^2-3*a*b*c*d+b^2*c^2)*x/d^3-1/3*b^2*(-3*a*d+b*c)*x^3/d^2+1/5*b^3*x^5/d-(-a*d+b*c)^3*arctan(x*d^(1/2)
/c^(1/2))/d^(7/2)/c^(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} \[ \frac {b x \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{d^3}-\frac {b^2 x^3 (b c-3 a d)}{3 d^2}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}}+\frac {b^3 x^5}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/(c + d*x^2),x]

[Out]

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

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Mathematica [A]  time = 0.06, size = 93, normalized size = 0.95 \[ \frac {b x \left (45 a^2 d^2+15 a b d \left (d x^2-3 c\right )+b^2 \left (15 c^2-5 c d x^2+3 d^2 x^4\right )\right )}{15 d^3}-\frac {(b c-a d)^3 \tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )}{\sqrt {c} d^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*(45*a^2*d^2 + 15*a*b*d*(-3*c + d*x^2) + b^2*(15*c^2 - 5*c*d*x^2 + 3*d^2*x^4)))/(15*d^3) - ((b*c - a*d)^3*
ArcTan[(Sqrt[d]*x)/Sqrt[c]])/(Sqrt[c]*d^(7/2))

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fricas [A]  time = 0.54, size = 290, normalized size = 2.96 \[ \left [\frac {6 \, b^{3} c d^{3} x^{5} - 10 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} x^{3} + 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-c d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-c d} x - c}{d x^{2} + c}\right ) + 30 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} x}{30 \, c d^{4}}, \frac {3 \, b^{3} c d^{3} x^{5} - 5 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3}\right )} x^{3} - 15 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {c d} \arctan \left (\frac {\sqrt {c d} x}{c}\right ) + 15 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3}\right )} x}{15 \, c d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.58, size = 130, normalized size = 1.33 \[ -\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{3} d^{4} x^{5} - 5 \, b^{3} c d^{3} x^{3} + 15 \, a b^{2} d^{4} x^{3} + 15 \, b^{3} c^{2} d^{2} x - 45 \, a b^{2} c d^{3} x + 45 \, a^{2} b d^{4} x}{15 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.02, size = 122, normalized size = 1.24 \[ -\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac {d x}{\sqrt {c d}}\right )}{\sqrt {c d} d^{3}} + \frac {3 \, b^{3} d^{2} x^{5} - 5 \, {\left (b^{3} c d - 3 \, a b^{2} d^{2}\right )} x^{3} + 15 \, {\left (b^{3} c^{2} - 3 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x}{15 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [B]  time = 4.87, size = 145, normalized size = 1.48 \[ x^3\,\left (\frac {a\,b^2}{d}-\frac {b^3\,c}{3\,d^2}\right )+x\,\left (\frac {3\,a^2\,b}{d}-\frac {c\,\left (\frac {3\,a\,b^2}{d}-\frac {b^3\,c}{d^2}\right )}{d}\right )+\frac {b^3\,x^5}{5\,d}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,x\,{\left (a\,d-b\,c\right )}^3}{\sqrt {c}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^3}{\sqrt {c}\,d^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/(c + d*x^2),x)

[Out]

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

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sympy [B]  time = 0.57, size = 238, normalized size = 2.43 \[ \frac {b^{3} x^{5}}{5 d} + x^{3} \left (\frac {a b^{2}}{d} - \frac {b^{3} c}{3 d^{2}}\right ) + x \left (\frac {3 a^{2} b}{d} - \frac {3 a b^{2} c}{d^{2}} + \frac {b^{3} c^{2}}{d^{3}}\right ) - \frac {\sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3} \log {\left (- \frac {c d^{3} \sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3} \log {\left (\frac {c d^{3} \sqrt {- \frac {1}{c d^{7}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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