3.27 \(\int \frac{(c+d x^2)^5}{(a+b x^2)^2} \, dx\)

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

[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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Rubi [A]  time = 0.162539, antiderivative size = 192, normalized size of antiderivative = 1., 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 = {390, 385, 205} \[ \frac{d^3 x^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{3 b^4}+\frac{d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}+\frac{(b c-a d)^4 (9 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{11/2}}+\frac{d^4 x^5 (5 b c-2 a d)}{5 b^3}+\frac{x (b c-a d)^5}{2 a b^5 \left (a+b x^2\right )}+\frac{d^5 x^7}{7 b^2} \]

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 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]

Rule 385

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 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]

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*}

Mathematica [A]  time = 0.0979393, size = 192, normalized size = 1. \[ \frac{d^3 x^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{3 b^4}+\frac{d^2 x \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}+\frac{(b c-a d)^4 (9 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{11/2}}+\frac{d^4 x^5 (5 b c-2 a d)}{5 b^3}+\frac{x (b c-a d)^5}{2 a b^5 \left (a+b x^2\right )}+\frac{d^5 x^7}{7 b^2} \]

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 [B]  time = 0.009, size = 402, normalized size = 2.1 \begin{align*}{\frac{{d}^{5}{x}^{7}}{7\,{b}^{2}}}-{\frac{2\,{d}^{5}{x}^{5}a}{5\,{b}^{3}}}+{\frac{{d}^{4}{x}^{5}c}{{b}^{2}}}+{\frac{{d}^{5}{x}^{3}{a}^{2}}{{b}^{4}}}-{\frac{10\,{d}^{4}{x}^{3}ac}{3\,{b}^{3}}}+{\frac{10\,{d}^{3}{x}^{3}{c}^{2}}{3\,{b}^{2}}}-4\,{\frac{{a}^{3}{d}^{5}x}{{b}^{5}}}+15\,{\frac{{a}^{2}c{d}^{4}x}{{b}^{4}}}-20\,{\frac{a{c}^{2}{d}^{3}x}{{b}^{3}}}+10\,{\frac{{c}^{3}{d}^{2}x}{{b}^{2}}}-{\frac{{a}^{4}x{d}^{5}}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{3}cx{d}^{4}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-5\,{\frac{{a}^{2}{c}^{2}x{d}^{3}}{{b}^{3} \left ( b{x}^{2}+a \right ) }}+5\,{\frac{a{c}^{3}x{d}^{2}}{{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{5\,x{c}^{4}d}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{x{c}^{5}}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{9\,{a}^{4}{d}^{5}}{2\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{35\,{a}^{3}c{d}^{4}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+25\,{\frac{{a}^{2}{c}^{2}{d}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-15\,{\frac{a{c}^{3}{d}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{5\,{c}^{4}d}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{5}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*d^5*x^7/b^2-2/5*d^5/b^3*x^5*a+d^4/b^2*x^5*c+d^5/b^4*x^3*a^2-10/3*d^4/b^3*x^3*a*c+10/3*d^3/b^2*x^3*c^2-4*d^
5/b^5*a^3*x+15*d^4/b^4*a^2*c*x-20*d^3/b^3*a*c^2*x+10*d^2/b^2*c^3*x-1/2/b^5*a^4*x/(b*x^2+a)*d^5+5/2/b^4*a^3*x/(
b*x^2+a)*c*d^4-5/b^3*a^2*x/(b*x^2+a)*c^2*d^3+5/b^2*a*x/(b*x^2+a)*c^3*d^2-5/2/b*x/(b*x^2+a)*c^4*d+1/2/a*x/(b*x^
2+a)*c^5+9/2/b^5*a^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d^5-35/2/b^4*a^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*
c*d^4+25/b^3*a^2/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^2*d^3-15/b^2*a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^3*
d^2+5/2/b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^4*d+1/2/a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.81095, size = 1675, normalized size = 8.72 \begin{align*} \text{result too large to display} \end{align*}

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]  time = 2.2884, size = 498, normalized size = 2.59 \begin{align*} - \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} \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} \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} \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} \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}} - \frac{x^{5} \left (2 a d^{5} - 5 b c d^{4}\right )}{5 b^{3}} + \frac{x^{3} \left (3 a^{2} d^{5} - 10 a b c d^{4} + 10 b^{2} c^{2} d^{3}\right )}{3 b^{4}} - \frac{x \left (4 a^{3} d^{5} - 15 a^{2} b c d^{4} + 20 a b^{2} c^{2} d^{3} - 10 b^{3} c^{3} d^{2}\right )}{b^{5}} \end{align*}

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*(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) - x**5*(2*a*d**5 -
5*b*c*d**4)/(5*b**3) + x**3*(3*a**2*d**5 - 10*a*b*c*d**4 + 10*b**2*c**2*d**3)/(3*b**4) - x*(4*a**3*d**5 - 15*a
**2*b*c*d**4 + 20*a*b**2*c**2*d**3 - 10*b**3*c**3*d**2)/b**5

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Giac [A]  time = 1.169, size = 413, normalized size = 2.15 \begin{align*} \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{align*}

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