3.1.36 \(\int \frac {(c+d x^2)^4}{(a+b x^2)^3} \, dx\) [36]

Optimal. Leaf size=160 \[ \frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}+\frac {(b c-a d)^3 (3 b c+13 a d) x}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{9/2}} \]

[Out]

d^3*(-3*a*d+4*b*c)*x/b^4+1/3*d^4*x^3/b^3+1/4*(-a*d+b*c)^4*x/a/b^4/(b*x^2+a)^2+1/8*(-a*d+b*c)^3*(13*a*d+3*b*c)*
x/a^2/b^4/(b*x^2+a)+1/8*(-a*d+b*c)^2*(35*a^2*d^2+10*a*b*c*d+3*b^2*c^2)*arctan(x*b^(1/2)/a^(1/2))/a^(5/2)/b^(9/
2)

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Rubi [A]
time = 0.13, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {398, 1171, 393, 211} \begin {gather*} \frac {x (b c-a d)^3 (13 a d+3 b c)}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^2 \left (35 a^2 d^2+10 a b c d+3 b^2 c^2\right )}{8 a^{5/2} b^{9/2}}+\frac {d^3 x (4 b c-3 a d)}{b^4}+\frac {x (b c-a d)^4}{4 a b^4 \left (a+b x^2\right )^2}+\frac {d^4 x^3}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4/(a + b*x^2)^3,x]

[Out]

(d^3*(4*b*c - 3*a*d)*x)/b^4 + (d^4*x^3)/(3*b^3) + ((b*c - a*d)^4*x)/(4*a*b^4*(a + b*x^2)^2) + ((b*c - a*d)^3*(
3*b*c + 13*a*d)*x)/(8*a^2*b^4*(a + b*x^2)) + ((b*c - a*d)^2*(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]])/(8*a^(5/2)*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]

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^4}{\left (a+b x^2\right )^3} \, dx &=\int \left (\frac {d^3 (4 b c-3 a d)}{b^4}+\frac {d^4 x^2}{b^3}+\frac {b^4 c^4-4 a^3 b c d^3+3 a^4 d^4+4 b d (b c-a d)^2 (b c+2 a d) x^2+6 b^2 d^2 (b c-a d)^2 x^4}{b^4 \left (a+b x^2\right )^3}\right ) \, dx\\ &=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {\int \frac {b^4 c^4-4 a^3 b c d^3+3 a^4 d^4+4 b d (b c-a d)^2 (b c+2 a d) x^2+6 b^2 d^2 (b c-a d)^2 x^4}{\left (a+b x^2\right )^3} \, dx}{b^4}\\ &=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}-\frac {\int \frac {-(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+11 a^2 d^2\right )-24 a b d^2 (b c-a d)^2 x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b^4}\\ &=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}+\frac {(b c-a d)^3 (3 b c+13 a d) x}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {\left ((b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 b^4}\\ &=\frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}+\frac {(b c-a d)^3 (3 b c+13 a d) x}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 160, normalized size = 1.00 \begin {gather*} \frac {d^3 (4 b c-3 a d) x}{b^4}+\frac {d^4 x^3}{3 b^3}+\frac {(b c-a d)^4 x}{4 a b^4 \left (a+b x^2\right )^2}+\frac {(b c-a d)^3 (3 b c+13 a d) x}{8 a^2 b^4 \left (a+b x^2\right )}+\frac {(b c-a d)^2 \left (3 b^2 c^2+10 a b c d+35 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^4/(a + b*x^2)^3,x]

[Out]

(d^3*(4*b*c - 3*a*d)*x)/b^4 + (d^4*x^3)/(3*b^3) + ((b*c - a*d)^4*x)/(4*a*b^4*(a + b*x^2)^2) + ((b*c - a*d)^3*(
3*b*c + 13*a*d)*x)/(8*a^2*b^4*(a + b*x^2)) + ((b*c - a*d)^2*(3*b^2*c^2 + 10*a*b*c*d + 35*a^2*d^2)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(9/2))

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Maple [A]
time = 0.11, size = 231, normalized size = 1.44

method	result	size
default	\(-\frac {d^{3} \left (-\frac {1}{3} b d \,x^{3}+3 a d x -4 b c x \right )}{b^{4}}+\frac {\frac {-\frac {b \left (13 a^{4} d^{4}-36 a^{3} b c \,d^{3}+30 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x^{3}}{8 a^{2}}-\frac {\left (11 a^{4} d^{4}-28 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d -5 b^{4} c^{4}\right ) x}{8 a}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (35 a^{4} d^{4}-60 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +3 b^{4} c^{4}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}}{b^{4}}\)	\(231\)
risch	\(\frac {d^{4} x^{3}}{3 b^{3}}-\frac {3 d^{4} a x}{b^{4}}+\frac {4 d^{3} c x}{b^{3}}+\frac {-\frac {b \left (13 a^{4} d^{4}-36 a^{3} b c \,d^{3}+30 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 b^{4} c^{4}\right ) x^{3}}{8 a^{2}}-\frac {\left (11 a^{4} d^{4}-28 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d -5 b^{4} c^{4}\right ) x}{8 a}}{b^{4} \left (b \,x^{2}+a \right )^{2}}-\frac {35 a^{2} \ln \left (b x +\sqrt {-a b}\right ) d^{4}}{16 b^{4} \sqrt {-a b}}+\frac {15 a \ln \left (b x +\sqrt {-a b}\right ) c \,d^{3}}{4 b^{3} \sqrt {-a b}}-\frac {9 \ln \left (b x +\sqrt {-a b}\right ) c^{2} d^{2}}{8 b^{2} \sqrt {-a b}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c^{3} d}{4 b \sqrt {-a b}\, a}-\frac {3 \ln \left (b x +\sqrt {-a b}\right ) c^{4}}{16 \sqrt {-a b}\, a^{2}}+\frac {35 a^{2} \ln \left (-b x +\sqrt {-a b}\right ) d^{4}}{16 b^{4} \sqrt {-a b}}-\frac {15 a \ln \left (-b x +\sqrt {-a b}\right ) c \,d^{3}}{4 b^{3} \sqrt {-a b}}+\frac {9 \ln \left (-b x +\sqrt {-a b}\right ) c^{2} d^{2}}{8 b^{2} \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c^{3} d}{4 b \sqrt {-a b}\, a}+\frac {3 \ln \left (-b x +\sqrt {-a b}\right ) c^{4}}{16 \sqrt {-a b}\, a^{2}}\)	\(443\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^4/(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

-d^3/b^4*(-1/3*b*d*x^3+3*a*d*x-4*b*c*x)+1/b^4*((-1/8*b*(13*a^4*d^4-36*a^3*b*c*d^3+30*a^2*b^2*c^2*d^2-4*a*b^3*c
^3*d-3*b^4*c^4)/a^2*x^3-1/8*(11*a^4*d^4-28*a^3*b*c*d^3+18*a^2*b^2*c^2*d^2+4*a*b^3*c^3*d-5*b^4*c^4)/a*x)/(b*x^2
+a)^2+1/8*(35*a^4*d^4-60*a^3*b*c*d^3+18*a^2*b^2*c^2*d^2+4*a*b^3*c^3*d+3*b^4*c^4)/a^2/(a*b)^(1/2)*arctan(b*x/(a
*b)^(1/2)))

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Maxima [A]
time = 0.49, size = 253, normalized size = 1.58 \begin {gather*} \frac {{\left (3 \, b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 30 \, a^{2} b^{3} c^{2} d^{2} + 36 \, a^{3} b^{2} c d^{3} - 13 \, a^{4} b d^{4}\right )} x^{3} + {\left (5 \, a b^{4} c^{4} - 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 28 \, a^{4} b c d^{3} - 11 \, a^{5} d^{4}\right )} x}{8 \, {\left (a^{2} b^{6} x^{4} + 2 \, a^{3} b^{5} x^{2} + a^{4} b^{4}\right )}} + \frac {b d^{4} x^{3} + 3 \, {\left (4 \, b c d^{3} - 3 \, a d^{4}\right )} x}{3 \, b^{4}} + \frac {{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} 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)^3,x, algorithm="maxima")

[Out]

1/8*((3*b^5*c^4 + 4*a*b^4*c^3*d - 30*a^2*b^3*c^2*d^2 + 36*a^3*b^2*c*d^3 - 13*a^4*b*d^4)*x^3 + (5*a*b^4*c^4 - 4
*a^2*b^3*c^3*d - 18*a^3*b^2*c^2*d^2 + 28*a^4*b*c*d^3 - 11*a^5*d^4)*x)/(a^2*b^6*x^4 + 2*a^3*b^5*x^2 + a^4*b^4)
+ 1/3*(b*d^4*x^3 + 3*(4*b*c*d^3 - 3*a*d^4)*x)/b^4 + 1/8*(3*b^4*c^4 + 4*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 60*a
^3*b*c*d^3 + 35*a^4*d^4)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^4)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (144) = 288\).
time = 0.49, size = 817, normalized size = 5.11 \begin {gather*} \left [\frac {16 \, a^{3} b^{4} d^{4} x^{7} + 16 \, {\left (12 \, a^{3} b^{4} c d^{3} - 7 \, a^{4} b^{3} d^{4}\right )} x^{5} + 2 \, {\left (9 \, a b^{6} c^{4} + 12 \, a^{2} b^{5} c^{3} d - 90 \, a^{3} b^{4} c^{2} d^{2} + 300 \, a^{4} b^{3} c d^{3} - 175 \, a^{5} b^{2} d^{4}\right )} x^{3} - 3 \, {\left (3 \, a^{2} b^{4} c^{4} + 4 \, a^{3} b^{3} c^{3} d + 18 \, a^{4} b^{2} c^{2} d^{2} - 60 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (3 \, b^{6} c^{4} + 4 \, a b^{5} c^{3} d + 18 \, a^{2} b^{4} c^{2} d^{2} - 60 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, a b^{5} c^{4} + 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 60 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} 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 ) + 6 \, {\left (5 \, a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d - 18 \, a^{4} b^{3} c^{2} d^{2} + 60 \, a^{5} b^{2} c d^{3} - 35 \, a^{6} b d^{4}\right )} x}{48 \, {\left (a^{3} b^{7} x^{4} + 2 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}, \frac {8 \, a^{3} b^{4} d^{4} x^{7} + 8 \, {\left (12 \, a^{3} b^{4} c d^{3} - 7 \, a^{4} b^{3} d^{4}\right )} x^{5} + {\left (9 \, a b^{6} c^{4} + 12 \, a^{2} b^{5} c^{3} d - 90 \, a^{3} b^{4} c^{2} d^{2} + 300 \, a^{4} b^{3} c d^{3} - 175 \, a^{5} b^{2} d^{4}\right )} x^{3} + 3 \, {\left (3 \, a^{2} b^{4} c^{4} + 4 \, a^{3} b^{3} c^{3} d + 18 \, a^{4} b^{2} c^{2} d^{2} - 60 \, a^{5} b c d^{3} + 35 \, a^{6} d^{4} + {\left (3 \, b^{6} c^{4} + 4 \, a b^{5} c^{3} d + 18 \, a^{2} b^{4} c^{2} d^{2} - 60 \, a^{3} b^{3} c d^{3} + 35 \, a^{4} b^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, a b^{5} c^{4} + 4 \, a^{2} b^{4} c^{3} d + 18 \, a^{3} b^{3} c^{2} d^{2} - 60 \, a^{4} b^{2} c d^{3} + 35 \, a^{5} b d^{4}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (5 \, a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d - 18 \, a^{4} b^{3} c^{2} d^{2} + 60 \, a^{5} b^{2} c d^{3} - 35 \, a^{6} b d^{4}\right )} x}{24 \, {\left (a^{3} b^{7} x^{4} + 2 \, a^{4} b^{6} x^{2} + a^{5} 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)^3,x, algorithm="fricas")

[Out]

[1/48*(16*a^3*b^4*d^4*x^7 + 16*(12*a^3*b^4*c*d^3 - 7*a^4*b^3*d^4)*x^5 + 2*(9*a*b^6*c^4 + 12*a^2*b^5*c^3*d - 90
*a^3*b^4*c^2*d^2 + 300*a^4*b^3*c*d^3 - 175*a^5*b^2*d^4)*x^3 - 3*(3*a^2*b^4*c^4 + 4*a^3*b^3*c^3*d + 18*a^4*b^2*
c^2*d^2 - 60*a^5*b*c*d^3 + 35*a^6*d^4 + (3*b^6*c^4 + 4*a*b^5*c^3*d + 18*a^2*b^4*c^2*d^2 - 60*a^3*b^3*c*d^3 + 3
5*a^4*b^2*d^4)*x^4 + 2*(3*a*b^5*c^4 + 4*a^2*b^4*c^3*d + 18*a^3*b^3*c^2*d^2 - 60*a^4*b^2*c*d^3 + 35*a^5*b*d^4)*
x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 6*(5*a^2*b^5*c^4 - 4*a^3*b^4*c^3*d - 18*a^4*b^
3*c^2*d^2 + 60*a^5*b^2*c*d^3 - 35*a^6*b*d^4)*x)/(a^3*b^7*x^4 + 2*a^4*b^6*x^2 + a^5*b^5), 1/24*(8*a^3*b^4*d^4*x
^7 + 8*(12*a^3*b^4*c*d^3 - 7*a^4*b^3*d^4)*x^5 + (9*a*b^6*c^4 + 12*a^2*b^5*c^3*d - 90*a^3*b^4*c^2*d^2 + 300*a^4
*b^3*c*d^3 - 175*a^5*b^2*d^4)*x^3 + 3*(3*a^2*b^4*c^4 + 4*a^3*b^3*c^3*d + 18*a^4*b^2*c^2*d^2 - 60*a^5*b*c*d^3 +
35*a^6*d^4 + (3*b^6*c^4 + 4*a*b^5*c^3*d + 18*a^2*b^4*c^2*d^2 - 60*a^3*b^3*c*d^3 + 35*a^4*b^2*d^4)*x^4 + 2*(3*
a*b^5*c^4 + 4*a^2*b^4*c^3*d + 18*a^3*b^3*c^2*d^2 - 60*a^4*b^2*c*d^3 + 35*a^5*b*d^4)*x^2)*sqrt(a*b)*arctan(sqrt
(a*b)*x/a) + 3*(5*a^2*b^5*c^4 - 4*a^3*b^4*c^3*d - 18*a^4*b^3*c^2*d^2 + 60*a^5*b^2*c*d^3 - 35*a^6*b*d^4)*x)/(a^
3*b^7*x^4 + 2*a^4*b^6*x^2 + a^5*b^5)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 515 vs. \(2 (150) = 300\).
time = 1.83, size = 515, normalized size = 3.22 \begin {gather*} x \left (- \frac {3 a d^{4}}{b^{4}} + \frac {4 c d^{3}}{b^{3}}\right ) - \frac {\sqrt {- \frac {1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \cdot \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right ) \log {\left (- \frac {a^{3} b^{4} \sqrt {- \frac {1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \cdot \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right )}{35 a^{4} d^{4} - 60 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} + 4 a b^{3} c^{3} d + 3 b^{4} c^{4}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \cdot \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right ) \log {\left (\frac {a^{3} b^{4} \sqrt {- \frac {1}{a^{5} b^{9}}} \left (a d - b c\right )^{2} \cdot \left (35 a^{2} d^{2} + 10 a b c d + 3 b^{2} c^{2}\right )}{35 a^{4} d^{4} - 60 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} + 4 a b^{3} c^{3} d + 3 b^{4} c^{4}} + x \right )}}{16} + \frac {x^{3} \left (- 13 a^{4} b d^{4} + 36 a^{3} b^{2} c d^{3} - 30 a^{2} b^{3} c^{2} d^{2} + 4 a b^{4} c^{3} d + 3 b^{5} c^{4}\right ) + x \left (- 11 a^{5} d^{4} + 28 a^{4} b c d^{3} - 18 a^{3} b^{2} c^{2} d^{2} - 4 a^{2} b^{3} c^{3} d + 5 a b^{4} c^{4}\right )}{8 a^{4} b^{4} + 16 a^{3} b^{5} x^{2} + 8 a^{2} b^{6} x^{4}} + \frac {d^{4} x^{3}}{3 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**4/(b*x**2+a)**3,x)

[Out]

x*(-3*a*d**4/b**4 + 4*c*d**3/b**3) - sqrt(-1/(a**5*b**9))*(a*d - b*c)**2*(35*a**2*d**2 + 10*a*b*c*d + 3*b**2*c
**2)*log(-a**3*b**4*sqrt(-1/(a**5*b**9))*(a*d - b*c)**2*(35*a**2*d**2 + 10*a*b*c*d + 3*b**2*c**2)/(35*a**4*d**
4 - 60*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 + 4*a*b**3*c**3*d + 3*b**4*c**4) + x)/16 + sqrt(-1/(a**5*b**9))*
(a*d - b*c)**2*(35*a**2*d**2 + 10*a*b*c*d + 3*b**2*c**2)*log(a**3*b**4*sqrt(-1/(a**5*b**9))*(a*d - b*c)**2*(35
*a**2*d**2 + 10*a*b*c*d + 3*b**2*c**2)/(35*a**4*d**4 - 60*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 + 4*a*b**3*c*
*3*d + 3*b**4*c**4) + x)/16 + (x**3*(-13*a**4*b*d**4 + 36*a**3*b**2*c*d**3 - 30*a**2*b**3*c**2*d**2 + 4*a*b**4
*c**3*d + 3*b**5*c**4) + x*(-11*a**5*d**4 + 28*a**4*b*c*d**3 - 18*a**3*b**2*c**2*d**2 - 4*a**2*b**3*c**3*d + 5
*a*b**4*c**4))/(8*a**4*b**4 + 16*a**3*b**5*x**2 + 8*a**2*b**6*x**4) + d**4*x**3/(3*b**3)

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Giac [A]
time = 1.43, size = 254, normalized size = 1.59 \begin {gather*} \frac {{\left (3 \, b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 60 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{4}} + \frac {3 \, b^{5} c^{4} x^{3} + 4 \, a b^{4} c^{3} d x^{3} - 30 \, a^{2} b^{3} c^{2} d^{2} x^{3} + 36 \, a^{3} b^{2} c d^{3} x^{3} - 13 \, a^{4} b d^{4} x^{3} + 5 \, a b^{4} c^{4} x - 4 \, a^{2} b^{3} c^{3} d x - 18 \, a^{3} b^{2} c^{2} d^{2} x + 28 \, a^{4} b c d^{3} x - 11 \, a^{5} d^{4} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b^{4}} + \frac {b^{6} d^{4} x^{3} + 12 \, b^{6} c d^{3} x - 9 \, a b^{5} d^{4} x}{3 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4/(b*x^2+a)^3,x, algorithm="giac")

[Out]

1/8*(3*b^4*c^4 + 4*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 35*a^4*d^4)*arctan(b*x/sqrt(a*b))/(sqrt
(a*b)*a^2*b^4) + 1/8*(3*b^5*c^4*x^3 + 4*a*b^4*c^3*d*x^3 - 30*a^2*b^3*c^2*d^2*x^3 + 36*a^3*b^2*c*d^3*x^3 - 13*a
^4*b*d^4*x^3 + 5*a*b^4*c^4*x - 4*a^2*b^3*c^3*d*x - 18*a^3*b^2*c^2*d^2*x + 28*a^4*b*c*d^3*x - 11*a^5*d^4*x)/((b
*x^2 + a)^2*a^2*b^4) + 1/3*(b^6*d^4*x^3 + 12*b^6*c*d^3*x - 9*a*b^5*d^4*x)/b^9

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Mupad [B]
time = 0.14, size = 318, normalized size = 1.99 \begin {gather*} \frac {d^4\,x^3}{3\,b^3}-x\,\left (\frac {3\,a\,d^4}{b^4}-\frac {4\,c\,d^3}{b^3}\right )-\frac {\frac {x\,\left (11\,a^4\,d^4-28\,a^3\,b\,c\,d^3+18\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d-5\,b^4\,c^4\right )}{8\,a}-\frac {x^3\,\left (-13\,a^4\,b\,d^4+36\,a^3\,b^2\,c\,d^3-30\,a^2\,b^3\,c^2\,d^2+4\,a\,b^4\,c^3\,d+3\,b^5\,c^4\right )}{8\,a^2}}{a^2\,b^4+2\,a\,b^5\,x^2+b^6\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2\,\left (35\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{\sqrt {a}\,\left (35\,a^4\,d^4-60\,a^3\,b\,c\,d^3+18\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+3\,b^4\,c^4\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (35\,a^2\,d^2+10\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,a^{5/2}\,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)^3,x)

[Out]

(d^4*x^3)/(3*b^3) - x*((3*a*d^4)/b^4 - (4*c*d^3)/b^3) - ((x*(11*a^4*d^4 - 5*b^4*c^4 + 18*a^2*b^2*c^2*d^2 + 4*a
*b^3*c^3*d - 28*a^3*b*c*d^3))/(8*a) - (x^3*(3*b^5*c^4 - 13*a^4*b*d^4 + 36*a^3*b^2*c*d^3 - 30*a^2*b^3*c^2*d^2 +
4*a*b^4*c^3*d))/(8*a^2))/(a^2*b^4 + b^6*x^4 + 2*a*b^5*x^2) + (atan((b^(1/2)*x*(a*d - b*c)^2*(35*a^2*d^2 + 3*b
^2*c^2 + 10*a*b*c*d))/(a^(1/2)*(35*a^4*d^4 + 3*b^4*c^4 + 18*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d - 60*a^3*b*c*d^3))
)*(a*d - b*c)^2*(35*a^2*d^2 + 3*b^2*c^2 + 10*a*b*c*d))/(8*a^(5/2)*b^(9/2))

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