3.1.27 \(\int \frac {(c+d x^2)^5}{(a+b x^2)^2} \, dx\) [27]

Optimal. Leaf size=192 \[ \frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^3}{3 b^4}+\frac {d^4 (5 b c-2 a d) x^5}{5 b^3}+\frac {d^5 x^7}{7 b^2}+\frac {(b c-a d)^5 x}{2 a b^5 \left (a+b x^2\right )}+\frac {(b c-a d)^4 (b c+9 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{11/2}} \]

[Out]

d^2*(-4*a^3*d^3+15*a^2*b*c*d^2-20*a*b^2*c^2*d+10*b^3*c^3)*x/b^5+1/3*d^3*(3*a^2*d^2-10*a*b*c*d+10*b^2*c^2)*x^3/
b^4+1/5*d^4*(-2*a*d+5*b*c)*x^5/b^3+1/7*d^5*x^7/b^2+1/2*(-a*d+b*c)^5*x/a/b^5/(b*x^2+a)+1/2*(-a*d+b*c)^4*(9*a*d+
b*c)*arctan(x*b^(1/2)/a^(1/2))/a^(3/2)/b^(11/2)

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Rubi [A]
time = 0.11, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {398, 393, 211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^4 (9 a d+b c)}{2 a^{3/2} b^{11/2}}+\frac {d^3 x^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{3 b^4}+\frac {d^2 x \left (-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}+\frac {x (b c-a d)^5}{2 a b^5 \left (a+b x^2\right )}+\frac {d^4 x^5 (5 b c-2 a d)}{5 b^3}+\frac {d^5 x^7}{7 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^5/(a + b*x^2)^2,x]

[Out]

(d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2
*d^2)*x^3)/(3*b^4) + (d^4*(5*b*c - 2*a*d)*x^5)/(5*b^3) + (d^5*x^7)/(7*b^2) + ((b*c - a*d)^5*x)/(2*a*b^5*(a + b
*x^2)) + ((b*c - a*d)^4*(b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*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]

Rubi steps

\begin {align*} \int \frac {\left (c+d x^2\right )^5}{\left (a+b x^2\right )^2} \, dx &=\int \left (\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right )}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^2}{b^4}+\frac {d^4 (5 b c-2 a d) x^4}{b^3}+\frac {d^5 x^6}{b^2}+\frac {(b c-a d)^4 (b c+4 a d)+5 b d (b c-a d)^4 x^2}{b^5 \left (a+b x^2\right )^2}\right ) \, dx\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^3}{3 b^4}+\frac {d^4 (5 b c-2 a d) x^5}{5 b^3}+\frac {d^5 x^7}{7 b^2}+\frac {\int \frac {(b c-a d)^4 (b c+4 a d)+5 b d (b c-a d)^4 x^2}{\left (a+b x^2\right )^2} \, dx}{b^5}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^3}{3 b^4}+\frac {d^4 (5 b c-2 a d) x^5}{5 b^3}+\frac {d^5 x^7}{7 b^2}+\frac {(b c-a d)^5 x}{2 a b^5 \left (a+b x^2\right )}+\frac {\left ((b c-a d)^4 (b c+9 a d)\right ) \int \frac {1}{a+b x^2} \, dx}{2 a b^5}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^3}{3 b^4}+\frac {d^4 (5 b c-2 a d) x^5}{5 b^3}+\frac {d^5 x^7}{7 b^2}+\frac {(b c-a d)^5 x}{2 a b^5 \left (a+b x^2\right )}+\frac {(b c-a d)^4 (b c+9 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 192, normalized size = 1.00 \begin {gather*} \frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^3}{3 b^4}+\frac {d^4 (5 b c-2 a d) x^5}{5 b^3}+\frac {d^5 x^7}{7 b^2}+\frac {(b c-a d)^5 x}{2 a b^5 \left (a+b x^2\right )}+\frac {(b c-a d)^4 (b c+9 a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2
*d^2)*x^3)/(3*b^4) + (d^4*(5*b*c - 2*a*d)*x^5)/(5*b^3) + (d^5*x^7)/(7*b^2) + ((b*c - a*d)^5*x)/(2*a*b^5*(a + b
*x^2)) + ((b*c - a*d)^4*(b*c + 9*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(11/2))

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Maple [A]
time = 0.09, size = 290, normalized size = 1.51

method	result	size
default	\(-\frac {d^{2} \left (-\frac {1}{7} d^{3} x^{7} b^{3}+\frac {2}{5} a \,b^{2} d^{3} x^{5}-b^{3} c \,d^{2} x^{5}-a^{2} b \,d^{3} x^{3}+\frac {10}{3} a \,b^{2} c \,d^{2} x^{3}-\frac {10}{3} b^{3} c^{2} d \,x^{3}+4 a^{3} d^{3} x -15 a^{2} b c \,d^{2} x +20 a \,b^{2} c^{2} d x -10 b^{3} c^{3} x \right )}{b^{5}}+\frac {-\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (9 a^{5} d^{5}-35 a^{4} b c \,d^{4}+50 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +b^{5} c^{5}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{5}}\)	\(290\)
risch	\(\frac {d^{5} x^{7}}{7 b^{2}}-\frac {2 d^{5} a \,x^{5}}{5 b^{3}}+\frac {d^{4} c \,x^{5}}{b^{2}}+\frac {d^{5} a^{2} x^{3}}{b^{4}}-\frac {10 d^{4} a c \,x^{3}}{3 b^{3}}+\frac {10 d^{3} c^{2} x^{3}}{3 b^{2}}-\frac {4 d^{5} a^{3} x}{b^{5}}+\frac {15 d^{4} a^{2} c x}{b^{4}}-\frac {20 d^{3} a \,c^{2} x}{b^{3}}+\frac {10 d^{2} c^{3} x}{b^{2}}-\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) x}{2 a \,b^{5} \left (b \,x^{2}+a \right )}-\frac {9 a^{4} \ln \left (b x +\sqrt {-a b}\right ) d^{5}}{4 b^{5} \sqrt {-a b}}+\frac {35 a^{3} \ln \left (b x +\sqrt {-a b}\right ) c \,d^{4}}{4 b^{4} \sqrt {-a b}}-\frac {25 a^{2} \ln \left (b x +\sqrt {-a b}\right ) c^{2} d^{3}}{2 b^{3} \sqrt {-a b}}+\frac {15 a \ln \left (b x +\sqrt {-a b}\right ) c^{3} d^{2}}{2 b^{2} \sqrt {-a b}}-\frac {5 \ln \left (b x +\sqrt {-a b}\right ) c^{4} d}{4 b \sqrt {-a b}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c^{5}}{4 \sqrt {-a b}\, a}+\frac {9 a^{4} \ln \left (-b x +\sqrt {-a b}\right ) d^{5}}{4 b^{5} \sqrt {-a b}}-\frac {35 a^{3} \ln \left (-b x +\sqrt {-a b}\right ) c \,d^{4}}{4 b^{4} \sqrt {-a b}}+\frac {25 a^{2} \ln \left (-b x +\sqrt {-a b}\right ) c^{2} d^{3}}{2 b^{3} \sqrt {-a b}}-\frac {15 a \ln \left (-b x +\sqrt {-a b}\right ) c^{3} d^{2}}{2 b^{2} \sqrt {-a b}}+\frac {5 \ln \left (-b x +\sqrt {-a b}\right ) c^{4} d}{4 b \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c^{5}}{4 \sqrt {-a b}\, a}\)	\(550\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d^2/b^5*(-1/7*d^3*x^7*b^3+2/5*a*b^2*d^3*x^5-b^3*c*d^2*x^5-a^2*b*d^3*x^3+10/3*a*b^2*c*d^2*x^3-10/3*b^3*c^2*d*x
^3+4*a^3*d^3*x-15*a^2*b*c*d^2*x+20*a*b^2*c^2*d*x-10*b^3*c^3*x)+1/b^5*(-1/2*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c
^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/a*x/(b*x^2+a)+1/2*(9*a^5*d^5-35*a^4*b*c*d^4+50*a^3*b^2*c^2*d^
3-30*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d+b^5*c^5)/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))

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Maxima [A]
time = 0.50, size = 294, normalized size = 1.53 \begin {gather*} \frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x}{2 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}} + \frac {15 \, b^{3} d^{5} x^{7} + 21 \, {\left (5 \, b^{3} c d^{4} - 2 \, a b^{2} d^{5}\right )} x^{5} + 35 \, {\left (10 \, b^{3} c^{2} d^{3} - 10 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} x^{3} + 105 \, {\left (10 \, b^{3} c^{3} d^{2} - 20 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b c d^{4} - 4 \, a^{3} d^{5}\right )} x}{105 \, b^{5}} + \frac {{\left (b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 30 \, a^{2} b^{3} c^{3} d^{2} + 50 \, a^{3} b^{2} c^{2} d^{3} - 35 \, a^{4} b c d^{4} + 9 \, a^{5} d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a 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)^2,x, algorithm="maxima")

[Out]

1/2*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*x/(a*b^6*x^2
+ a^2*b^5) + 1/105*(15*b^3*d^5*x^7 + 21*(5*b^3*c*d^4 - 2*a*b^2*d^5)*x^5 + 35*(10*b^3*c^2*d^3 - 10*a*b^2*c*d^4
+ 3*a^2*b*d^5)*x^3 + 105*(10*b^3*c^3*d^2 - 20*a*b^2*c^2*d^3 + 15*a^2*b*c*d^4 - 4*a^3*d^5)*x)/b^5 + 1/2*(b^5*c
^5 + 5*a*b^4*c^4*d - 30*a^2*b^3*c^3*d^2 + 50*a^3*b^2*c^2*d^3 - 35*a^4*b*c*d^4 + 9*a^5*d^5)*arctan(b*x/sqrt(a*b
))/(sqrt(a*b)*a*b^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (174) = 348\).
time = 0.79, size = 810, normalized size = 4.22 \begin {gather*} \left [\frac {60 \, a^{2} b^{5} d^{5} x^{9} + 12 \, {\left (35 \, a^{2} b^{5} c d^{4} - 9 \, a^{3} b^{4} d^{5}\right )} x^{7} + 28 \, {\left (50 \, a^{2} b^{5} c^{2} d^{3} - 35 \, a^{3} b^{4} c d^{4} + 9 \, a^{4} b^{3} d^{5}\right )} x^{5} + 140 \, {\left (30 \, a^{2} b^{5} c^{3} d^{2} - 50 \, a^{3} b^{4} c^{2} d^{3} + 35 \, a^{4} b^{3} c d^{4} - 9 \, a^{5} b^{2} d^{5}\right )} x^{3} - 105 \, {\left (a b^{5} c^{5} + 5 \, a^{2} b^{4} c^{4} d - 30 \, a^{3} b^{3} c^{3} d^{2} + 50 \, a^{4} b^{2} c^{2} d^{3} - 35 \, a^{5} b c d^{4} + 9 \, a^{6} d^{5} + {\left (b^{6} c^{5} + 5 \, a b^{5} c^{4} d - 30 \, a^{2} b^{4} c^{3} d^{2} + 50 \, a^{3} b^{3} c^{2} d^{3} - 35 \, a^{4} b^{2} c d^{4} + 9 \, a^{5} 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 ) + 210 \, {\left (a b^{6} c^{5} - 5 \, a^{2} b^{5} c^{4} d + 30 \, a^{3} b^{4} c^{3} d^{2} - 50 \, a^{4} b^{3} c^{2} d^{3} + 35 \, a^{5} b^{2} c d^{4} - 9 \, a^{6} b d^{5}\right )} x}{420 \, {\left (a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}, \frac {30 \, a^{2} b^{5} d^{5} x^{9} + 6 \, {\left (35 \, a^{2} b^{5} c d^{4} - 9 \, a^{3} b^{4} d^{5}\right )} x^{7} + 14 \, {\left (50 \, a^{2} b^{5} c^{2} d^{3} - 35 \, a^{3} b^{4} c d^{4} + 9 \, a^{4} b^{3} d^{5}\right )} x^{5} + 70 \, {\left (30 \, a^{2} b^{5} c^{3} d^{2} - 50 \, a^{3} b^{4} c^{2} d^{3} + 35 \, a^{4} b^{3} c d^{4} - 9 \, a^{5} b^{2} d^{5}\right )} x^{3} + 105 \, {\left (a b^{5} c^{5} + 5 \, a^{2} b^{4} c^{4} d - 30 \, a^{3} b^{3} c^{3} d^{2} + 50 \, a^{4} b^{2} c^{2} d^{3} - 35 \, a^{5} b c d^{4} + 9 \, a^{6} d^{5} + {\left (b^{6} c^{5} + 5 \, a b^{5} c^{4} d - 30 \, a^{2} b^{4} c^{3} d^{2} + 50 \, a^{3} b^{3} c^{2} d^{3} - 35 \, a^{4} b^{2} c d^{4} + 9 \, a^{5} b d^{5}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 105 \, {\left (a b^{6} c^{5} - 5 \, a^{2} b^{5} c^{4} d + 30 \, a^{3} b^{4} c^{3} d^{2} - 50 \, a^{4} b^{3} c^{2} d^{3} + 35 \, a^{5} b^{2} c d^{4} - 9 \, a^{6} b d^{5}\right )} x}{210 \, {\left (a^{2} b^{7} x^{2} + a^{3} 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)^2,x, algorithm="fricas")

[Out]

[1/420*(60*a^2*b^5*d^5*x^9 + 12*(35*a^2*b^5*c*d^4 - 9*a^3*b^4*d^5)*x^7 + 28*(50*a^2*b^5*c^2*d^3 - 35*a^3*b^4*c
*d^4 + 9*a^4*b^3*d^5)*x^5 + 140*(30*a^2*b^5*c^3*d^2 - 50*a^3*b^4*c^2*d^3 + 35*a^4*b^3*c*d^4 - 9*a^5*b^2*d^5)*x
^3 - 105*(a*b^5*c^5 + 5*a^2*b^4*c^4*d - 30*a^3*b^3*c^3*d^2 + 50*a^4*b^2*c^2*d^3 - 35*a^5*b*c*d^4 + 9*a^6*d^5 +
(b^6*c^5 + 5*a*b^5*c^4*d - 30*a^2*b^4*c^3*d^2 + 50*a^3*b^3*c^2*d^3 - 35*a^4*b^2*c*d^4 + 9*a^5*b*d^5)*x^2)*sqr
t(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 210*(a*b^6*c^5 - 5*a^2*b^5*c^4*d + 30*a^3*b^4*c^3*d^2
- 50*a^4*b^3*c^2*d^3 + 35*a^5*b^2*c*d^4 - 9*a^6*b*d^5)*x)/(a^2*b^7*x^2 + a^3*b^6), 1/210*(30*a^2*b^5*d^5*x^9 +
6*(35*a^2*b^5*c*d^4 - 9*a^3*b^4*d^5)*x^7 + 14*(50*a^2*b^5*c^2*d^3 - 35*a^3*b^4*c*d^4 + 9*a^4*b^3*d^5)*x^5 + 7
0*(30*a^2*b^5*c^3*d^2 - 50*a^3*b^4*c^2*d^3 + 35*a^4*b^3*c*d^4 - 9*a^5*b^2*d^5)*x^3 + 105*(a*b^5*c^5 + 5*a^2*b^
4*c^4*d - 30*a^3*b^3*c^3*d^2 + 50*a^4*b^2*c^2*d^3 - 35*a^5*b*c*d^4 + 9*a^6*d^5 + (b^6*c^5 + 5*a*b^5*c^4*d - 30
*a^2*b^4*c^3*d^2 + 50*a^3*b^3*c^2*d^3 - 35*a^4*b^2*c*d^4 + 9*a^5*b*d^5)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) +
105*(a*b^6*c^5 - 5*a^2*b^5*c^4*d + 30*a^3*b^4*c^3*d^2 - 50*a^4*b^3*c^2*d^3 + 35*a^5*b^2*c*d^4 - 9*a^6*b*d^5)*
x)/(a^2*b^7*x^2 + a^3*b^6)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (185) = 370\).
time = 1.11, size = 502, normalized size = 2.61 \begin {gather*} x^{5} \left (- \frac {2 a d^{5}}{5 b^{3}} + \frac {c d^{4}}{b^{2}}\right ) + x^{3} \left (\frac {a^{2} d^{5}}{b^{4}} - \frac {10 a c d^{4}}{3 b^{3}} + \frac {10 c^{2} d^{3}}{3 b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{5}}{b^{5}} + \frac {15 a^{2} c d^{4}}{b^{4}} - \frac {20 a c^{2} d^{3}}{b^{3}} + \frac {10 c^{3} d^{2}}{b^{2}}\right ) + \frac {x \left (- a^{5} d^{5} + 5 a^{4} b c d^{4} - 10 a^{3} b^{2} c^{2} d^{3} + 10 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d + b^{5} c^{5}\right )}{2 a^{2} b^{5} + 2 a b^{6} x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b^{11}}} \left (a d - b c\right )^{4} \cdot \left (9 a d + b c\right ) \log {\left (- \frac {a^{2} b^{5} \sqrt {- \frac {1}{a^{3} b^{11}}} \left (a d - b c\right )^{4} \cdot \left (9 a d + b c\right )}{9 a^{5} d^{5} - 35 a^{4} b c d^{4} + 50 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d + b^{5} c^{5}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b^{11}}} \left (a d - b c\right )^{4} \cdot \left (9 a d + b c\right ) \log {\left (\frac {a^{2} b^{5} \sqrt {- \frac {1}{a^{3} b^{11}}} \left (a d - b c\right )^{4} \cdot \left (9 a d + b c\right )}{9 a^{5} d^{5} - 35 a^{4} b c d^{4} + 50 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d + b^{5} c^{5}} + x \right )}}{4} + \frac {d^{5} x^{7}}{7 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5*(-2*a*d**5/(5*b**3) + c*d**4/b**2) + x**3*(a**2*d**5/b**4 - 10*a*c*d**4/(3*b**3) + 10*c**2*d**3/(3*b**2))
+ x*(-4*a**3*d**5/b**5 + 15*a**2*c*d**4/b**4 - 20*a*c**2*d**3/b**3 + 10*c**3*d**2/b**2) + x*(-a**5*d**5 + 5*a
**4*b*c*d**4 - 10*a**3*b**2*c**2*d**3 + 10*a**2*b**3*c**3*d**2 - 5*a*b**4*c**4*d + b**5*c**5)/(2*a**2*b**5 + 2
*a*b**6*x**2) - sqrt(-1/(a**3*b**11))*(a*d - b*c)**4*(9*a*d + b*c)*log(-a**2*b**5*sqrt(-1/(a**3*b**11))*(a*d -
b*c)**4*(9*a*d + b*c)/(9*a**5*d**5 - 35*a**4*b*c*d**4 + 50*a**3*b**2*c**2*d**3 - 30*a**2*b**3*c**3*d**2 + 5*a
*b**4*c**4*d + b**5*c**5) + x)/4 + sqrt(-1/(a**3*b**11))*(a*d - b*c)**4*(9*a*d + b*c)*log(a**2*b**5*sqrt(-1/(a
**3*b**11))*(a*d - b*c)**4*(9*a*d + b*c)/(9*a**5*d**5 - 35*a**4*b*c*d**4 + 50*a**3*b**2*c**2*d**3 - 30*a**2*b*
*3*c**3*d**2 + 5*a*b**4*c**4*d + b**5*c**5) + x)/4 + d**5*x**7/(7*b**2)

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Giac [A]
time = 0.85, size = 306, normalized size = 1.59 \begin {gather*} \frac {{\left (b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 30 \, a^{2} b^{3} c^{3} d^{2} + 50 \, a^{3} b^{2} c^{2} d^{3} - 35 \, a^{4} b c d^{4} + 9 \, a^{5} d^{5}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{5}} + \frac {b^{5} c^{5} x - 5 \, a b^{4} c^{4} d x + 10 \, a^{2} b^{3} c^{3} d^{2} x - 10 \, a^{3} b^{2} c^{2} d^{3} x + 5 \, a^{4} b c d^{4} x - a^{5} d^{5} x}{2 \, {\left (b x^{2} + a\right )} a b^{5}} + \frac {15 \, b^{12} d^{5} x^{7} + 105 \, b^{12} c d^{4} x^{5} - 42 \, a b^{11} d^{5} x^{5} + 350 \, b^{12} c^{2} d^{3} x^{3} - 350 \, a b^{11} c d^{4} x^{3} + 105 \, a^{2} b^{10} d^{5} x^{3} + 1050 \, b^{12} c^{3} d^{2} x - 2100 \, a b^{11} c^{2} d^{3} x + 1575 \, a^{2} b^{10} c d^{4} x - 420 \, a^{3} b^{9} d^{5} x}{105 \, b^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^5*c^5 + 5*a*b^4*c^4*d - 30*a^2*b^3*c^3*d^2 + 50*a^3*b^2*c^2*d^3 - 35*a^4*b*c*d^4 + 9*a^5*d^5)*arctan(b*
x/sqrt(a*b))/(sqrt(a*b)*a*b^5) + 1/2*(b^5*c^5*x - 5*a*b^4*c^4*d*x + 10*a^2*b^3*c^3*d^2*x - 10*a^3*b^2*c^2*d^3*
x + 5*a^4*b*c*d^4*x - a^5*d^5*x)/((b*x^2 + a)*a*b^5) + 1/105*(15*b^12*d^5*x^7 + 105*b^12*c*d^4*x^5 - 42*a*b^11
*d^5*x^5 + 350*b^12*c^2*d^3*x^3 - 350*a*b^11*c*d^4*x^3 + 105*a^2*b^10*d^5*x^3 + 1050*b^12*c^3*d^2*x - 2100*a*b
^11*c^2*d^3*x + 1575*a^2*b^10*c*d^4*x - 420*a^3*b^9*d^5*x)/b^14

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Mupad [B]
time = 5.02, size = 386, normalized size = 2.01 \begin {gather*} x\,\left (\frac {10\,c^3\,d^2}{b^2}-\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^5}{b^3}-\frac {5\,c\,d^4}{b^2}\right )}{b}-\frac {a^2\,d^5}{b^4}+\frac {10\,c^2\,d^3}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {2\,a\,d^5}{b^3}-\frac {5\,c\,d^4}{b^2}\right )}{b^2}\right )-x^5\,\left (\frac {2\,a\,d^5}{5\,b^3}-\frac {c\,d^4}{b^2}\right )+x^3\,\left (\frac {2\,a\,\left (\frac {2\,a\,d^5}{b^3}-\frac {5\,c\,d^4}{b^2}\right )}{3\,b}-\frac {a^2\,d^5}{3\,b^4}+\frac {10\,c^2\,d^3}{3\,b^2}\right )+\frac {d^5\,x^7}{7\,b^2}-\frac {x\,\left (a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5\right )}{2\,a\,\left (b^6\,x^2+a\,b^5\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^4\,\left (9\,a\,d+b\,c\right )}{\sqrt {a}\,\left (9\,a^5\,d^5-35\,a^4\,b\,c\,d^4+50\,a^3\,b^2\,c^2\,d^3-30\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d+b^5\,c^5\right )}\right )\,{\left (a\,d-b\,c\right )}^4\,\left (9\,a\,d+b\,c\right )}{2\,a^{3/2}\,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)^2,x)

[Out]

x*((10*c^3*d^2)/b^2 - (2*a*((2*a*((2*a*d^5)/b^3 - (5*c*d^4)/b^2))/b - (a^2*d^5)/b^4 + (10*c^2*d^3)/b^2))/b + (
a^2*((2*a*d^5)/b^3 - (5*c*d^4)/b^2))/b^2) - x^5*((2*a*d^5)/(5*b^3) - (c*d^4)/b^2) + x^3*((2*a*((2*a*d^5)/b^3 -
(5*c*d^4)/b^2))/(3*b) - (a^2*d^5)/(3*b^4) + (10*c^2*d^3)/(3*b^2)) + (d^5*x^7)/(7*b^2) - (x*(a^5*d^5 - b^5*c^5
- 10*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^2*d^3 + 5*a*b^4*c^4*d - 5*a^4*b*c*d^4))/(2*a*(a*b^5 + b^6*x^2)) + (atan((
b^(1/2)*x*(a*d - b*c)^4*(9*a*d + b*c))/(a^(1/2)*(9*a^5*d^5 + b^5*c^5 - 30*a^2*b^3*c^3*d^2 + 50*a^3*b^2*c^2*d^3
+ 5*a*b^4*c^4*d - 35*a^4*b*c*d^4)))*(a*d - b*c)^4*(9*a*d + b*c))/(2*a^(3/2)*b^(11/2))

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