3.1.35 \(\int \frac {(c+d x^2)^5}{(a+b x^2)^3} \, dx\) [35]

Optimal. Leaf size=196 \[ \frac {d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac {d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac {d^5 x^5}{5 b^3}+\frac {(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}+\frac {(b c-a d)^4 (3 b c+17 a d) x}{8 a^2 b^5 \left (a+b x^2\right )}+\frac {(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{11/2}} \]

[Out]

d^3*(6*a^2*d^2-15*a*b*c*d+10*b^2*c^2)*x/b^5+1/3*d^4*(-3*a*d+5*b*c)*x^3/b^4+1/5*d^5*x^5/b^3+1/4*(-a*d+b*c)^5*x/
a/b^5/(b*x^2+a)^2+1/8*(-a*d+b*c)^4*(17*a*d+3*b*c)*x/a^2/b^5/(b*x^2+a)+1/8*(-a*d+b*c)^3*(63*a^2*d^2+14*a*b*c*d+
3*b^2*c^2)*arctan(x*b^(1/2)/a^(1/2))/a^(5/2)/b^(11/2)

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Rubi [A]
time = 0.15, antiderivative size = 196, 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 (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac {d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3}{8 a^{5/2} b^{11/2}}+\frac {x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac {d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac {d^5 x^5}{5 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^5/(a + b*x^2)^3,x]

[Out]

(d^3*(10*b^2*c^2 - 15*a*b*c*d + 6*a^2*d^2)*x)/b^5 + (d^4*(5*b*c - 3*a*d)*x^3)/(3*b^4) + (d^5*x^5)/(5*b^3) + ((
b*c - a*d)^5*x)/(4*a*b^5*(a + b*x^2)^2) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*x)/(8*a^2*b^5*(a + b*x^2)) + ((b*c -
a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(11/2))

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 )^5}{\left (a+b x^2\right )^3} \, dx &=\int \left (\frac {d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right )}{b^5}+\frac {d^4 (5 b c-3 a d) x^2}{b^4}+\frac {d^5 x^4}{b^3}+\frac {(b c-a d)^3 \left (b^2 c^2+3 a b c d+6 a^2 d^2\right )+5 b d (b c-a d)^3 (b c+3 a d) x^2+10 b^2 d^2 (b c-a d)^3 x^4}{b^5 \left (a+b x^2\right )^3}\right ) \, dx\\ &=\frac {d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac {d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac {d^5 x^5}{5 b^3}+\frac {\int \frac {(b c-a d)^3 \left (b^2 c^2+3 a b c d+6 a^2 d^2\right )+5 b d (b c-a d)^3 (b c+3 a d) x^2+10 b^2 d^2 (b c-a d)^3 x^4}{\left (a+b x^2\right )^3} \, dx}{b^5}\\ &=\frac {d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac {d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac {d^5 x^5}{5 b^3}+\frac {(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}-\frac {\int \frac {-(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+23 a^2 d^2\right )-40 a b d^2 (b c-a d)^3 x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b^5}\\ &=\frac {d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac {d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac {d^5 x^5}{5 b^3}+\frac {(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}+\frac {(b c-a d)^4 (3 b c+17 a d) x}{8 a^2 b^5 \left (a+b x^2\right )}+\frac {\left ((b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right )\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^2 b^5}\\ &=\frac {d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac {d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac {d^5 x^5}{5 b^3}+\frac {(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}+\frac {(b c-a d)^4 (3 b c+17 a d) x}{8 a^2 b^5 \left (a+b x^2\right )}+\frac {(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 196, normalized size = 1.00 \begin {gather*} \frac {d^3 \left (10 b^2 c^2-15 a b c d+6 a^2 d^2\right ) x}{b^5}+\frac {d^4 (5 b c-3 a d) x^3}{3 b^4}+\frac {d^5 x^5}{5 b^3}+\frac {(b c-a d)^5 x}{4 a b^5 \left (a+b x^2\right )^2}+\frac {(b c-a d)^4 (3 b c+17 a d) x}{8 a^2 b^5 \left (a+b x^2\right )}+\frac {(b c-a d)^3 \left (3 b^2 c^2+14 a b c d+63 a^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^5/(a + b*x^2)^3,x]

[Out]

(d^3*(10*b^2*c^2 - 15*a*b*c*d + 6*a^2*d^2)*x)/b^5 + (d^4*(5*b*c - 3*a*d)*x^3)/(3*b^4) + (d^5*x^5)/(5*b^3) + ((
b*c - a*d)^5*x)/(4*a*b^5*(a + b*x^2)^2) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*x)/(8*a^2*b^5*(a + b*x^2)) + ((b*c -
a*d)^3*(3*b^2*c^2 + 14*a*b*c*d + 63*a^2*d^2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(5/2)*b^(11/2))

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Maple [A]
time = 0.10, size = 311, normalized size = 1.59

method	result	size
default	\(\frac {d^{3} \left (\frac {1}{5} b^{2} x^{5} d^{2}-a b \,d^{2} x^{3}+\frac {5}{3} b^{2} c d \,x^{3}+6 a^{2} d^{2} x -15 a b c d x +10 b^{2} c^{2} x \right )}{b^{5}}-\frac {\frac {-\frac {b \left (17 a^{5} d^{5}-65 a^{4} b c \,d^{4}+90 a^{3} b^{2} c^{2} d^{3}-50 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right ) x^{3}}{8 a^{2}}-\frac {5 \left (3 a^{5} d^{5}-11 a^{4} b c \,d^{4}+14 a^{3} b^{2} c^{2} d^{3}-6 a^{2} b^{3} c^{3} d^{2}-a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x}{8 a}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (63 a^{5} d^{5}-175 a^{4} b c \,d^{4}+150 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}-5 a \,b^{4} c^{4} d -3 b^{5} c^{5}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 a^{2} \sqrt {a b}}}{b^{5}}\)	\(311\)
risch	\(\frac {d^{5} x^{5}}{5 b^{3}}-\frac {d^{5} a \,x^{3}}{b^{4}}+\frac {5 d^{4} c \,x^{3}}{3 b^{3}}+\frac {6 d^{5} a^{2} x}{b^{5}}-\frac {15 d^{4} a c x}{b^{4}}+\frac {10 d^{3} c^{2} x}{b^{3}}+\frac {\frac {b \left (17 a^{5} d^{5}-65 a^{4} b c \,d^{4}+90 a^{3} b^{2} c^{2} d^{3}-50 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right ) x^{3}}{8 a^{2}}+\frac {5 \left (3 a^{5} d^{5}-11 a^{4} b c \,d^{4}+14 a^{3} b^{2} c^{2} d^{3}-6 a^{2} b^{3} c^{3} d^{2}-a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x}{8 a}}{b^{5} \left (b \,x^{2}+a \right )^{2}}-\frac {63 a^{3} \ln \left (b x -\sqrt {-a b}\right ) d^{5}}{16 b^{5} \sqrt {-a b}}+\frac {175 a^{2} \ln \left (b x -\sqrt {-a b}\right ) c \,d^{4}}{16 b^{4} \sqrt {-a b}}-\frac {75 a \ln \left (b x -\sqrt {-a b}\right ) c^{2} d^{3}}{8 b^{3} \sqrt {-a b}}+\frac {15 \ln \left (b x -\sqrt {-a b}\right ) c^{3} d^{2}}{8 b^{2} \sqrt {-a b}}+\frac {5 \ln \left (b x -\sqrt {-a b}\right ) c^{4} d}{16 b \sqrt {-a b}\, a}+\frac {3 \ln \left (b x -\sqrt {-a b}\right ) c^{5}}{16 \sqrt {-a b}\, a^{2}}+\frac {63 a^{3} \ln \left (-b x -\sqrt {-a b}\right ) d^{5}}{16 b^{5} \sqrt {-a b}}-\frac {175 a^{2} \ln \left (-b x -\sqrt {-a b}\right ) c \,d^{4}}{16 b^{4} \sqrt {-a b}}+\frac {75 a \ln \left (-b x -\sqrt {-a b}\right ) c^{2} d^{3}}{8 b^{3} \sqrt {-a b}}-\frac {15 \ln \left (-b x -\sqrt {-a b}\right ) c^{3} d^{2}}{8 b^{2} \sqrt {-a b}}-\frac {5 \ln \left (-b x -\sqrt {-a b}\right ) c^{4} d}{16 b \sqrt {-a b}\, a}-\frac {3 \ln \left (-b x -\sqrt {-a b}\right ) c^{5}}{16 \sqrt {-a b}\, a^{2}}\)	\(596\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^5/(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

d^3/b^5*(1/5*b^2*x^5*d^2-a*b*d^2*x^3+5/3*b^2*c*d*x^3+6*a^2*d^2*x-15*a*b*c*d*x+10*b^2*c^2*x)-1/b^5*((-1/8*b*(17
*a^5*d^5-65*a^4*b*c*d^4+90*a^3*b^2*c^2*d^3-50*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d+3*b^5*c^5)/a^2*x^3-5/8*(3*a^5*d^5-
11*a^4*b*c*d^4+14*a^3*b^2*c^2*d^3-6*a^2*b^3*c^3*d^2-a*b^4*c^4*d+b^5*c^5)/a*x)/(b*x^2+a)^2+1/8*(63*a^5*d^5-175*
a^4*b*c*d^4+150*a^3*b^2*c^2*d^3-30*a^2*b^3*c^3*d^2-5*a*b^4*c^4*d-3*b^5*c^5)/a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(
1/2)))

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Maxima [A]
time = 0.52, size = 334, normalized size = 1.70 \begin {gather*} \frac {{\left (3 \, b^{6} c^{5} + 5 \, a b^{5} c^{4} d - 50 \, a^{2} b^{4} c^{3} d^{2} + 90 \, a^{3} b^{3} c^{2} d^{3} - 65 \, a^{4} b^{2} c d^{4} + 17 \, a^{5} b d^{5}\right )} x^{3} + 5 \, {\left (a b^{5} c^{5} - a^{2} b^{4} c^{4} d - 6 \, a^{3} b^{3} c^{3} d^{2} + 14 \, a^{4} b^{2} c^{2} d^{3} - 11 \, a^{5} b c d^{4} + 3 \, a^{6} d^{5}\right )} x}{8 \, {\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} + \frac {3 \, b^{2} d^{5} x^{5} + 5 \, {\left (5 \, b^{2} c d^{4} - 3 \, a b d^{5}\right )} x^{3} + 15 \, {\left (10 \, b^{2} c^{2} d^{3} - 15 \, a b c d^{4} + 6 \, a^{2} d^{5}\right )} x}{15 \, b^{5}} + \frac {{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^5/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

1/8*((3*b^6*c^5 + 5*a*b^5*c^4*d - 50*a^2*b^4*c^3*d^2 + 90*a^3*b^3*c^2*d^3 - 65*a^4*b^2*c*d^4 + 17*a^5*b*d^5)*x
^3 + 5*(a*b^5*c^5 - a^2*b^4*c^4*d - 6*a^3*b^3*c^3*d^2 + 14*a^4*b^2*c^2*d^3 - 11*a^5*b*c*d^4 + 3*a^6*d^5)*x)/(a
^2*b^7*x^4 + 2*a^3*b^6*x^2 + a^4*b^5) + 1/15*(3*b^2*d^5*x^5 + 5*(5*b^2*c*d^4 - 3*a*b*d^5)*x^3 + 15*(10*b^2*c^2
*d^3 - 15*a*b*c*d^4 + 6*a^2*d^5)*x)/b^5 + 1/8*(3*b^5*c^5 + 5*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 150*a^3*b^2*c^
2*d^3 + 175*a^4*b*c*d^4 - 63*a^5*d^5)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 512 vs. \(2 (178) = 356\).
time = 0.53, size = 1044, normalized size = 5.33 \begin {gather*} \left [\frac {48 \, a^{3} b^{5} d^{5} x^{9} + 16 \, {\left (25 \, a^{3} b^{5} c d^{4} - 9 \, a^{4} b^{4} d^{5}\right )} x^{7} + 16 \, {\left (150 \, a^{3} b^{5} c^{2} d^{3} - 175 \, a^{4} b^{4} c d^{4} + 63 \, a^{5} b^{3} d^{5}\right )} x^{5} + 10 \, {\left (9 \, a b^{7} c^{5} + 15 \, a^{2} b^{6} c^{4} d - 150 \, a^{3} b^{5} c^{3} d^{2} + 750 \, a^{4} b^{4} c^{2} d^{3} - 875 \, a^{5} b^{3} c d^{4} + 315 \, a^{6} b^{2} d^{5}\right )} x^{3} + 15 \, {\left (3 \, a^{2} b^{5} c^{5} + 5 \, a^{3} b^{4} c^{4} d + 30 \, a^{4} b^{3} c^{3} d^{2} - 150 \, a^{5} b^{2} c^{2} d^{3} + 175 \, a^{6} b c d^{4} - 63 \, a^{7} d^{5} + {\left (3 \, b^{7} c^{5} + 5 \, a b^{6} c^{4} d + 30 \, a^{2} b^{5} c^{3} d^{2} - 150 \, a^{3} b^{4} c^{2} d^{3} + 175 \, a^{4} b^{3} c d^{4} - 63 \, a^{5} b^{2} d^{5}\right )} x^{4} + 2 \, {\left (3 \, a b^{6} c^{5} + 5 \, a^{2} b^{5} c^{4} d + 30 \, a^{3} b^{4} c^{3} d^{2} - 150 \, a^{4} b^{3} c^{2} d^{3} + 175 \, a^{5} b^{2} c d^{4} - 63 \, a^{6} b d^{5}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 30 \, {\left (5 \, a^{2} b^{6} c^{5} - 5 \, a^{3} b^{5} c^{4} d - 30 \, a^{4} b^{4} c^{3} d^{2} + 150 \, a^{5} b^{3} c^{2} d^{3} - 175 \, a^{6} b^{2} c d^{4} + 63 \, a^{7} b d^{5}\right )} x}{240 \, {\left (a^{3} b^{8} x^{4} + 2 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}, \frac {24 \, a^{3} b^{5} d^{5} x^{9} + 8 \, {\left (25 \, a^{3} b^{5} c d^{4} - 9 \, a^{4} b^{4} d^{5}\right )} x^{7} + 8 \, {\left (150 \, a^{3} b^{5} c^{2} d^{3} - 175 \, a^{4} b^{4} c d^{4} + 63 \, a^{5} b^{3} d^{5}\right )} x^{5} + 5 \, {\left (9 \, a b^{7} c^{5} + 15 \, a^{2} b^{6} c^{4} d - 150 \, a^{3} b^{5} c^{3} d^{2} + 750 \, a^{4} b^{4} c^{2} d^{3} - 875 \, a^{5} b^{3} c d^{4} + 315 \, a^{6} b^{2} d^{5}\right )} x^{3} + 15 \, {\left (3 \, a^{2} b^{5} c^{5} + 5 \, a^{3} b^{4} c^{4} d + 30 \, a^{4} b^{3} c^{3} d^{2} - 150 \, a^{5} b^{2} c^{2} d^{3} + 175 \, a^{6} b c d^{4} - 63 \, a^{7} d^{5} + {\left (3 \, b^{7} c^{5} + 5 \, a b^{6} c^{4} d + 30 \, a^{2} b^{5} c^{3} d^{2} - 150 \, a^{3} b^{4} c^{2} d^{3} + 175 \, a^{4} b^{3} c d^{4} - 63 \, a^{5} b^{2} d^{5}\right )} x^{4} + 2 \, {\left (3 \, a b^{6} c^{5} + 5 \, a^{2} b^{5} c^{4} d + 30 \, a^{3} b^{4} c^{3} d^{2} - 150 \, a^{4} b^{3} c^{2} d^{3} + 175 \, a^{5} b^{2} c d^{4} - 63 \, a^{6} b d^{5}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (5 \, a^{2} b^{6} c^{5} - 5 \, a^{3} b^{5} c^{4} d - 30 \, a^{4} b^{4} c^{3} d^{2} + 150 \, a^{5} b^{3} c^{2} d^{3} - 175 \, a^{6} b^{2} c d^{4} + 63 \, a^{7} b d^{5}\right )} x}{120 \, {\left (a^{3} b^{8} x^{4} + 2 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^5/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/240*(48*a^3*b^5*d^5*x^9 + 16*(25*a^3*b^5*c*d^4 - 9*a^4*b^4*d^5)*x^7 + 16*(150*a^3*b^5*c^2*d^3 - 175*a^4*b^4
*c*d^4 + 63*a^5*b^3*d^5)*x^5 + 10*(9*a*b^7*c^5 + 15*a^2*b^6*c^4*d - 150*a^3*b^5*c^3*d^2 + 750*a^4*b^4*c^2*d^3
- 875*a^5*b^3*c*d^4 + 315*a^6*b^2*d^5)*x^3 + 15*(3*a^2*b^5*c^5 + 5*a^3*b^4*c^4*d + 30*a^4*b^3*c^3*d^2 - 150*a^
5*b^2*c^2*d^3 + 175*a^6*b*c*d^4 - 63*a^7*d^5 + (3*b^7*c^5 + 5*a*b^6*c^4*d + 30*a^2*b^5*c^3*d^2 - 150*a^3*b^4*c
^2*d^3 + 175*a^4*b^3*c*d^4 - 63*a^5*b^2*d^5)*x^4 + 2*(3*a*b^6*c^5 + 5*a^2*b^5*c^4*d + 30*a^3*b^4*c^3*d^2 - 150
*a^4*b^3*c^2*d^3 + 175*a^5*b^2*c*d^4 - 63*a^6*b*d^5)*x^2)*sqrt(-a*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 +
a)) + 30*(5*a^2*b^6*c^5 - 5*a^3*b^5*c^4*d - 30*a^4*b^4*c^3*d^2 + 150*a^5*b^3*c^2*d^3 - 175*a^6*b^2*c*d^4 + 63
*a^7*b*d^5)*x)/(a^3*b^8*x^4 + 2*a^4*b^7*x^2 + a^5*b^6), 1/120*(24*a^3*b^5*d^5*x^9 + 8*(25*a^3*b^5*c*d^4 - 9*a^
4*b^4*d^5)*x^7 + 8*(150*a^3*b^5*c^2*d^3 - 175*a^4*b^4*c*d^4 + 63*a^5*b^3*d^5)*x^5 + 5*(9*a*b^7*c^5 + 15*a^2*b^
6*c^4*d - 150*a^3*b^5*c^3*d^2 + 750*a^4*b^4*c^2*d^3 - 875*a^5*b^3*c*d^4 + 315*a^6*b^2*d^5)*x^3 + 15*(3*a^2*b^5
*c^5 + 5*a^3*b^4*c^4*d + 30*a^4*b^3*c^3*d^2 - 150*a^5*b^2*c^2*d^3 + 175*a^6*b*c*d^4 - 63*a^7*d^5 + (3*b^7*c^5
+ 5*a*b^6*c^4*d + 30*a^2*b^5*c^3*d^2 - 150*a^3*b^4*c^2*d^3 + 175*a^4*b^3*c*d^4 - 63*a^5*b^2*d^5)*x^4 + 2*(3*a*
b^6*c^5 + 5*a^2*b^5*c^4*d + 30*a^3*b^4*c^3*d^2 - 150*a^4*b^3*c^2*d^3 + 175*a^5*b^2*c*d^4 - 63*a^6*b*d^5)*x^2)*
sqrt(a*b)*arctan(sqrt(a*b)*x/a) + 15*(5*a^2*b^6*c^5 - 5*a^3*b^5*c^4*d - 30*a^4*b^4*c^3*d^2 + 150*a^5*b^3*c^2*d
^3 - 175*a^6*b^2*c*d^4 + 63*a^7*b*d^5)*x)/(a^3*b^8*x^4 + 2*a^4*b^7*x^2 + a^5*b^6)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (187) = 374\).
time = 7.27, size = 615, normalized size = 3.14 \begin {gather*} x^{3} \left (- \frac {a d^{5}}{b^{4}} + \frac {5 c d^{4}}{3 b^{3}}\right ) + x \left (\frac {6 a^{2} d^{5}}{b^{5}} - \frac {15 a c d^{4}}{b^{4}} + \frac {10 c^{2} d^{3}}{b^{3}}\right ) + \frac {\sqrt {- \frac {1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \cdot \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \log {\left (- \frac {a^{3} b^{5} \sqrt {- \frac {1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \cdot \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{63 a^{5} d^{5} - 175 a^{4} b c d^{4} + 150 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d - 3 b^{5} c^{5}} + x \right )}}{16} - \frac {\sqrt {- \frac {1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \cdot \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \log {\left (\frac {a^{3} b^{5} \sqrt {- \frac {1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \cdot \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{63 a^{5} d^{5} - 175 a^{4} b c d^{4} + 150 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d - 3 b^{5} c^{5}} + x \right )}}{16} + \frac {x^{3} \cdot \left (17 a^{5} b d^{5} - 65 a^{4} b^{2} c d^{4} + 90 a^{3} b^{3} c^{2} d^{3} - 50 a^{2} b^{4} c^{3} d^{2} + 5 a b^{5} c^{4} d + 3 b^{6} c^{5}\right ) + x \left (15 a^{6} d^{5} - 55 a^{5} b c d^{4} + 70 a^{4} b^{2} c^{2} d^{3} - 30 a^{3} b^{3} c^{3} d^{2} - 5 a^{2} b^{4} c^{4} d + 5 a b^{5} c^{5}\right )}{8 a^{4} b^{5} + 16 a^{3} b^{6} x^{2} + 8 a^{2} b^{7} x^{4}} + \frac {d^{5} x^{5}}{5 b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**5/(b*x**2+a)**3,x)

[Out]

x**3*(-a*d**5/b**4 + 5*c*d**4/(3*b**3)) + x*(6*a**2*d**5/b**5 - 15*a*c*d**4/b**4 + 10*c**2*d**3/b**3) + sqrt(-
1/(a**5*b**11))*(a*d - b*c)**3*(63*a**2*d**2 + 14*a*b*c*d + 3*b**2*c**2)*log(-a**3*b**5*sqrt(-1/(a**5*b**11))*
(a*d - b*c)**3*(63*a**2*d**2 + 14*a*b*c*d + 3*b**2*c**2)/(63*a**5*d**5 - 175*a**4*b*c*d**4 + 150*a**3*b**2*c**
2*d**3 - 30*a**2*b**3*c**3*d**2 - 5*a*b**4*c**4*d - 3*b**5*c**5) + x)/16 - sqrt(-1/(a**5*b**11))*(a*d - b*c)**
3*(63*a**2*d**2 + 14*a*b*c*d + 3*b**2*c**2)*log(a**3*b**5*sqrt(-1/(a**5*b**11))*(a*d - b*c)**3*(63*a**2*d**2 +
14*a*b*c*d + 3*b**2*c**2)/(63*a**5*d**5 - 175*a**4*b*c*d**4 + 150*a**3*b**2*c**2*d**3 - 30*a**2*b**3*c**3*d**
2 - 5*a*b**4*c**4*d - 3*b**5*c**5) + x)/16 + (x**3*(17*a**5*b*d**5 - 65*a**4*b**2*c*d**4 + 90*a**3*b**3*c**2*d
**3 - 50*a**2*b**4*c**3*d**2 + 5*a*b**5*c**4*d + 3*b**6*c**5) + x*(15*a**6*d**5 - 55*a**5*b*c*d**4 + 70*a**4*b
**2*c**2*d**3 - 30*a**3*b**3*c**3*d**2 - 5*a**2*b**4*c**4*d + 5*a*b**5*c**5))/(8*a**4*b**5 + 16*a**3*b**6*x**2
+ 8*a**2*b**7*x**4) + d**5*x**5/(5*b**3)

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Giac [A]
time = 1.49, size = 340, normalized size = 1.73 \begin {gather*} \frac {{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2} b^{5}} + \frac {3 \, b^{6} c^{5} x^{3} + 5 \, a b^{5} c^{4} d x^{3} - 50 \, a^{2} b^{4} c^{3} d^{2} x^{3} + 90 \, a^{3} b^{3} c^{2} d^{3} x^{3} - 65 \, a^{4} b^{2} c d^{4} x^{3} + 17 \, a^{5} b d^{5} x^{3} + 5 \, a b^{5} c^{5} x - 5 \, a^{2} b^{4} c^{4} d x - 30 \, a^{3} b^{3} c^{3} d^{2} x + 70 \, a^{4} b^{2} c^{2} d^{3} x - 55 \, a^{5} b c d^{4} x + 15 \, a^{6} d^{5} x}{8 \, {\left (b x^{2} + a\right )}^{2} a^{2} b^{5}} + \frac {3 \, b^{12} d^{5} x^{5} + 25 \, b^{12} c d^{4} x^{3} - 15 \, a b^{11} d^{5} x^{3} + 150 \, b^{12} c^{2} d^{3} x - 225 \, a b^{11} c d^{4} x + 90 \, a^{2} b^{10} d^{5} x}{15 \, b^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^5/(b*x^2+a)^3,x, algorithm="giac")

[Out]

1/8*(3*b^5*c^5 + 5*a*b^4*c^4*d + 30*a^2*b^3*c^3*d^2 - 150*a^3*b^2*c^2*d^3 + 175*a^4*b*c*d^4 - 63*a^5*d^5)*arct
an(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^5) + 1/8*(3*b^6*c^5*x^3 + 5*a*b^5*c^4*d*x^3 - 50*a^2*b^4*c^3*d^2*x^3 + 90*a
^3*b^3*c^2*d^3*x^3 - 65*a^4*b^2*c*d^4*x^3 + 17*a^5*b*d^5*x^3 + 5*a*b^5*c^5*x - 5*a^2*b^4*c^4*d*x - 30*a^3*b^3*
c^3*d^2*x + 70*a^4*b^2*c^2*d^3*x - 55*a^5*b*c*d^4*x + 15*a^6*d^5*x)/((b*x^2 + a)^2*a^2*b^5) + 1/15*(3*b^12*d^5
*x^5 + 25*b^12*c*d^4*x^3 - 15*a*b^11*d^5*x^3 + 150*b^12*c^2*d^3*x - 225*a*b^11*c*d^4*x + 90*a^2*b^10*d^5*x)/b^
15

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Mupad [B]
time = 5.02, size = 409, normalized size = 2.09 \begin {gather*} \frac {\frac {5\,x\,\left (3\,a^5\,d^5-11\,a^4\,b\,c\,d^4+14\,a^3\,b^2\,c^2\,d^3-6\,a^2\,b^3\,c^3\,d^2-a\,b^4\,c^4\,d+b^5\,c^5\right )}{8\,a}+\frac {x^3\,\left (17\,a^5\,b\,d^5-65\,a^4\,b^2\,c\,d^4+90\,a^3\,b^3\,c^2\,d^3-50\,a^2\,b^4\,c^3\,d^2+5\,a\,b^5\,c^4\,d+3\,b^6\,c^5\right )}{8\,a^2}}{a^2\,b^5+2\,a\,b^6\,x^2+b^7\,x^4}-x^3\,\left (\frac {a\,d^5}{b^4}-\frac {5\,c\,d^4}{3\,b^3}\right )+x\,\left (\frac {3\,a\,\left (\frac {3\,a\,d^5}{b^4}-\frac {5\,c\,d^4}{b^3}\right )}{b}-\frac {3\,a^2\,d^5}{b^5}+\frac {10\,c^2\,d^3}{b^3}\right )+\frac {d^5\,x^5}{5\,b^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^3\,\left (63\,a^2\,d^2+14\,a\,b\,c\,d+3\,b^2\,c^2\right )}{\sqrt {a}\,\left (-63\,a^5\,d^5+175\,a^4\,b\,c\,d^4-150\,a^3\,b^2\,c^2\,d^3+30\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d+3\,b^5\,c^5\right )}\right )\,{\left (a\,d-b\,c\right )}^3\,\left (63\,a^2\,d^2+14\,a\,b\,c\,d+3\,b^2\,c^2\right )}{8\,a^{5/2}\,b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)^5/(a + b*x^2)^3,x)

[Out]

((5*x*(3*a^5*d^5 + b^5*c^5 - 6*a^2*b^3*c^3*d^2 + 14*a^3*b^2*c^2*d^3 - a*b^4*c^4*d - 11*a^4*b*c*d^4))/(8*a) + (
x^3*(3*b^6*c^5 + 17*a^5*b*d^5 - 65*a^4*b^2*c*d^4 - 50*a^2*b^4*c^3*d^2 + 90*a^3*b^3*c^2*d^3 + 5*a*b^5*c^4*d))/(
8*a^2))/(a^2*b^5 + b^7*x^4 + 2*a*b^6*x^2) - x^3*((a*d^5)/b^4 - (5*c*d^4)/(3*b^3)) + x*((3*a*((3*a*d^5)/b^4 - (
5*c*d^4)/b^3))/b - (3*a^2*d^5)/b^5 + (10*c^2*d^3)/b^3) + (d^5*x^5)/(5*b^3) + (atan((b^(1/2)*x*(a*d - b*c)^3*(6
3*a^2*d^2 + 3*b^2*c^2 + 14*a*b*c*d))/(a^(1/2)*(3*b^5*c^5 - 63*a^5*d^5 + 30*a^2*b^3*c^3*d^2 - 150*a^3*b^2*c^2*d
^3 + 5*a*b^4*c^4*d + 175*a^4*b*c*d^4)))*(a*d - b*c)^3*(63*a^2*d^2 + 3*b^2*c^2 + 14*a*b*c*d))/(8*a^(5/2)*b^(11/
2))

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