3.20 \(\int \frac {(c+d x^2)^4}{a+b x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 142, 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} \[ \frac {d^2 x^3 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right )}{3 b^3}+\frac {d x (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right )}{b^4}+\frac {d^3 x^5 (4 b c-a d)}{5 b^2}+\frac {(b c-a d)^4 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{9/2}}+\frac {d^4 x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4/(a + b*x^2),x]

[Out]

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

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

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Mathematica [A]  time = 0.09, size = 136, normalized size = 0.96 \[ \frac {d x \left (-105 a^3 d^3+35 a^2 b d^2 \left (12 c+d x^2\right )-7 a b^2 d \left (90 c^2+20 c d x^2+3 d^2 x^4\right )+3 b^3 \left (140 c^3+70 c^2 d x^2+28 c d^2 x^4+5 d^3 x^6\right )\right )}{105 b^4}+\frac {(b c-a d)^4 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*(-105*a^3*d^3 + 35*a^2*b*d^2*(12*c + d*x^2) - 7*a*b^2*d*(90*c^2 + 20*c*d*x^2 + 3*d^2*x^4) + 3*b^3*(140*c^
3 + 70*c^2*d*x^2 + 28*c*d^2*x^4 + 5*d^3*x^6)))/(105*b^4) + ((b*c - a*d)^4*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a
]*b^(9/2))

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fricas [A]  time = 0.65, size = 428, normalized size = 3.01 \[ \left [\frac {30 \, a b^{4} d^{4} x^{7} + 42 \, {\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 70 \, {\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} - 105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 210 \, {\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{210 \, a b^{5}}, \frac {15 \, a b^{4} d^{4} x^{7} + 21 \, {\left (4 \, a b^{4} c d^{3} - a^{2} b^{3} d^{4}\right )} x^{5} + 35 \, {\left (6 \, a b^{4} c^{2} d^{2} - 4 \, a^{2} b^{3} c d^{3} + a^{3} b^{2} d^{4}\right )} x^{3} + 105 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 105 \, {\left (4 \, a b^{4} c^{3} d - 6 \, a^{2} b^{3} c^{2} d^{2} + 4 \, a^{3} b^{2} c d^{3} - a^{4} b d^{4}\right )} x}{105 \, a b^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(30*a*b^4*d^4*x^7 + 42*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^5 + 70*(6*a*b^4*c^2*d^2 - 4*a^2*b^3*c*d^3 + a^3*
b^2*d^4)*x^3 - 105*(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)*sqrt(-a*b)*log((b*x
^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 210*(4*a*b^4*c^3*d - 6*a^2*b^3*c^2*d^2 + 4*a^3*b^2*c*d^3 - a^4*b*d^4)*
x)/(a*b^5), 1/105*(15*a*b^4*d^4*x^7 + 21*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^5 + 35*(6*a*b^4*c^2*d^2 - 4*a^2*b^3*c
*d^3 + a^3*b^2*d^4)*x^3 + 105*(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)*sqrt(a*b
)*arctan(sqrt(a*b)*x/a) + 105*(4*a*b^4*c^3*d - 6*a^2*b^3*c^2*d^2 + 4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)/(a*b^5)]

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giac [A]  time = 0.58, size = 198, normalized size = 1.39 \[ \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{6} d^{4} x^{7} + 84 \, b^{6} c d^{3} x^{5} - 21 \, a b^{5} d^{4} x^{5} + 210 \, b^{6} c^{2} d^{2} x^{3} - 140 \, a b^{5} c d^{3} x^{3} + 35 \, a^{2} b^{4} d^{4} x^{3} + 420 \, b^{6} c^{3} d x - 630 \, a b^{5} c^{2} d^{2} x + 420 \, a^{2} b^{4} c d^{3} x - 105 \, a^{3} b^{3} d^{4} x}{105 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4)
+ 1/105*(15*b^6*d^4*x^7 + 84*b^6*c*d^3*x^5 - 21*a*b^5*d^4*x^5 + 210*b^6*c^2*d^2*x^3 - 140*a*b^5*c*d^3*x^3 + 35
*a^2*b^4*d^4*x^3 + 420*b^6*c^3*d*x - 630*a*b^5*c^2*d^2*x + 420*a^2*b^4*c*d^3*x - 105*a^3*b^3*d^4*x)/b^7

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maple [A]  time = 0.01, size = 246, normalized size = 1.73 \[ \frac {d^{4} x^{7}}{7 b}-\frac {a \,d^{4} x^{5}}{5 b^{2}}+\frac {4 c \,d^{3} x^{5}}{5 b}+\frac {a^{2} d^{4} x^{3}}{3 b^{3}}-\frac {4 a c \,d^{3} x^{3}}{3 b^{2}}+\frac {2 c^{2} d^{2} x^{3}}{b}+\frac {a^{4} d^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {4 a^{3} c \,d^{3} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {6 a^{2} c^{2} d^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {4 a \,c^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}+\frac {c^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {a^{3} d^{4} x}{b^{4}}+\frac {4 a^{2} c \,d^{3} x}{b^{3}}-\frac {6 a \,c^{2} d^{2} x}{b^{2}}+\frac {4 c^{3} d x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.03, size = 187, normalized size = 1.32 \[ \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{3} d^{4} x^{7} + 21 \, {\left (4 \, b^{3} c d^{3} - a b^{2} d^{4}\right )} x^{5} + 35 \, {\left (6 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{3} + 105 \, {\left (4 \, b^{3} c^{3} d - 6 \, a b^{2} c^{2} d^{2} + 4 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x}{105 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^4)
+ 1/105*(15*b^3*d^4*x^7 + 21*(4*b^3*c*d^3 - a*b^2*d^4)*x^5 + 35*(6*b^3*c^2*d^2 - 4*a*b^2*c*d^3 + a^2*b*d^4)*x^
3 + 105*(4*b^3*c^3*d - 6*a*b^2*c^2*d^2 + 4*a^2*b*c*d^3 - a^3*d^4)*x)/b^4

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mupad [B]  time = 4.86, size = 216, normalized size = 1.52 \[ x\,\left (\frac {4\,c^3\,d}{b}-\frac {a\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{b}+\frac {6\,c^2\,d^2}{b}\right )}{b}\right )-x^5\,\left (\frac {a\,d^4}{5\,b^2}-\frac {4\,c\,d^3}{5\,b}\right )+x^3\,\left (\frac {a\,\left (\frac {a\,d^4}{b^2}-\frac {4\,c\,d^3}{b}\right )}{3\,b}+\frac {2\,c^2\,d^2}{b}\right )+\frac {d^4\,x^7}{7\,b}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^4}{\sqrt {a}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )\,{\left (a\,d-b\,c\right )}^4}{\sqrt {a}\,b^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^4/(a + b*x^2),x)

[Out]

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

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sympy [B]  time = 0.76, size = 326, normalized size = 2.30 \[ x^{5} \left (- \frac {a d^{4}}{5 b^{2}} + \frac {4 c d^{3}}{5 b}\right ) + x^{3} \left (\frac {a^{2} d^{4}}{3 b^{3}} - \frac {4 a c d^{3}}{3 b^{2}} + \frac {2 c^{2} d^{2}}{b}\right ) + x \left (- \frac {a^{3} d^{4}}{b^{4}} + \frac {4 a^{2} c d^{3}}{b^{3}} - \frac {6 a c^{2} d^{2}}{b^{2}} + \frac {4 c^{3} d}{b}\right ) - \frac {\sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4} \log {\left (- \frac {a b^{4} \sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4} \log {\left (\frac {a b^{4} \sqrt {- \frac {1}{a b^{9}}} \left (a d - b c\right )^{4}}{a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}} + x \right )}}{2} + \frac {d^{4} x^{7}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5*(-a*d**4/(5*b**2) + 4*c*d**3/(5*b)) + x**3*(a**2*d**4/(3*b**3) - 4*a*c*d**3/(3*b**2) + 2*c**2*d**2/b) + x
*(-a**3*d**4/b**4 + 4*a**2*c*d**3/b**3 - 6*a*c**2*d**2/b**2 + 4*c**3*d/b) - sqrt(-1/(a*b**9))*(a*d - b*c)**4*l
og(-a*b**4*sqrt(-1/(a*b**9))*(a*d - b*c)**4/(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) + x)/2 + sqrt(-1/(a*b**9))*(a*d - b*c)**4*log(a*b**4*sqrt(-1/(a*b**9))*(a*d - b*c)**4/(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) + x)/2 + d**4*x**7/(7*b)

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