3.35 \(\int \frac{\left (c+d x^2\right )^5}{\left (a+b x^2\right )^3} \, dx\)

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

[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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Rubi [A]  time = 0.476364, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{d^5 x^5}{5 b^3} \]

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))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{3} \left (6 a^{2} d^{2} - 15 a b c d + 10 b^{2} c^{2}\right ) \int \frac{1}{b^{5}}\, dx + \frac{d^{5} x^{5}}{5 b^{3}} - \frac{d^{4} x^{3} \left (3 a d - 5 b c\right )}{3 b^{4}} - \frac{x \left (a d - b c\right )^{5}}{4 a b^{5} \left (a + b x^{2}\right )^{2}} + \frac{x \left (a d - b c\right )^{4} \left (17 a d + 3 b c\right )}{8 a^{2} b^{5} \left (a + b x^{2}\right )} - \frac{\left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} b^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*(6*a**2*d**2 - 15*a*b*c*d + 10*b**2*c**2)*Integral(b**(-5), x) + d**5*x**5/
(5*b**3) - d**4*x**3*(3*a*d - 5*b*c)/(3*b**4) - x*(a*d - b*c)**5/(4*a*b**5*(a +
b*x**2)**2) + x*(a*d - b*c)**4*(17*a*d + 3*b*c)/(8*a**2*b**5*(a + b*x**2)) - (a*
d - b*c)**3*(63*a**2*d**2 + 14*a*b*c*d + 3*b**2*c**2)*atan(sqrt(b)*x/sqrt(a))/(8
*a**(5/2)*b**(11/2))

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Mathematica [A]  time = 0.201818, size = 196, normalized size = 1. \[ \frac{x (17 a d+3 b c) (b c-a d)^4}{8 a^2 b^5 \left (a+b x^2\right )}+\frac{d^3 x \left (6 a^2 d^2-15 a b c d+10 b^2 c^2\right )}{b^5}+\frac{\left (63 a^2 d^2+14 a b c d+3 b^2 c^2\right ) (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{11/2}}+\frac{x (b c-a d)^5}{4 a b^5 \left (a+b x^2\right )^2}+\frac{d^4 x^3 (5 b c-3 a d)}{3 b^4}+\frac{d^5 x^5}{5 b^3} \]

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 [B]  time = 0.019, size = 484, normalized size = 2.5 \[{\frac{{d}^{5}{x}^{5}}{5\,{b}^{3}}}-{\frac{{d}^{5}{x}^{3}a}{{b}^{4}}}+{\frac{5\,{d}^{4}{x}^{3}c}{3\,{b}^{3}}}+6\,{\frac{{a}^{2}{d}^{5}x}{{b}^{5}}}-15\,{\frac{ac{d}^{4}x}{{b}^{4}}}+10\,{\frac{{c}^{2}{d}^{3}x}{{b}^{3}}}+{\frac{17\,{a}^{3}{x}^{3}{d}^{5}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{65\,{a}^{2}{x}^{3}c{d}^{4}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{45\,a{x}^{3}{c}^{2}{d}^{3}}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{25\,{x}^{3}{c}^{3}{d}^{2}}{4\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,{x}^{3}{c}^{4}d}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,b{x}^{3}{c}^{5}}{8\, \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}+{\frac{15\,{a}^{4}x{d}^{5}}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{55\,{a}^{3}cx{d}^{4}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{a}^{2}{c}^{2}x{d}^{3}}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,a{c}^{3}x{d}^{2}}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,x{c}^{4}d}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,x{c}^{5}}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{63\,{a}^{3}{d}^{5}}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{175\,{a}^{2}c{d}^{4}}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{75\,a{c}^{2}{d}^{3}}{4\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{15\,{c}^{3}{d}^{2}}{4\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{c}^{4}d}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{5}}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^5/(b*x^2+a)^3,x)

[Out]

1/5*d^5*x^5/b^3-d^5/b^4*x^3*a+5/3*d^4/b^3*x^3*c+6*d^5/b^5*a^2*x-15*d^4/b^4*a*c*x
+10*d^3/b^3*c^2*x+17/8/b^4/(b*x^2+a)^2*a^3*x^3*d^5-65/8/b^3/(b*x^2+a)^2*a^2*x^3*
c*d^4+45/4/b^2/(b*x^2+a)^2*a*x^3*c^2*d^3-25/4/b/(b*x^2+a)^2*x^3*c^3*d^2+5/8/(b*x
^2+a)^2/a*x^3*c^4*d+3/8*b/(b*x^2+a)^2/a^2*x^3*c^5+15/8/b^5/(b*x^2+a)^2*a^4*x*d^5
-55/8/b^4/(b*x^2+a)^2*a^3*x*c*d^4+35/4/b^3/(b*x^2+a)^2*a^2*x*c^2*d^3-15/4/b^2/(b
*x^2+a)^2*a*x*c^3*d^2-5/8/b/(b*x^2+a)^2*x*c^4*d+5/8/(b*x^2+a)^2/a*x*c^5-63/8/b^5
*a^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d^5+175/8/b^4*a^2/(a*b)^(1/2)*arctan(x*
b/(a*b)^(1/2))*c*d^4-75/4/b^3*a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^2*d^3+15/4
/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^3*d^2+5/8/b/a/(a*b)^(1/2)*arctan(x*b/
(a*b)^(1/2))*c^4*d+3/8/a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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]

Exception raised: ValueError

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Fricas [A]  time = 0.219103, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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*(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)*log(-(2*a*b*x - (b*x^2 - a)*sqrt(-a*b))/(b*x^2 +
a)) - 2*(24*a^2*b^4*d^5*x^9 + 8*(25*a^2*b^4*c*d^4 - 9*a^3*b^3*d^5)*x^7 + 8*(150
*a^2*b^4*c^2*d^3 - 175*a^3*b^3*c*d^4 + 63*a^4*b^2*d^5)*x^5 + 5*(9*b^6*c^5 + 15*a
*b^5*c^4*d - 150*a^2*b^4*c^3*d^2 + 750*a^3*b^3*c^2*d^3 - 875*a^4*b^2*c*d^4 + 315
*a^5*b*d^5)*x^3 + 15*(5*a*b^5*c^5 - 5*a^2*b^4*c^4*d - 30*a^3*b^3*c^3*d^2 + 150*a
^4*b^2*c^2*d^3 - 175*a^5*b*c*d^4 + 63*a^6*d^5)*x)*sqrt(-a*b))/((a^2*b^7*x^4 + 2*
a^3*b^6*x^2 + a^4*b^5)*sqrt(-a*b)), 1/120*(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)*arctan(sqrt(a*
b)*x/a) + (24*a^2*b^4*d^5*x^9 + 8*(25*a^2*b^4*c*d^4 - 9*a^3*b^3*d^5)*x^7 + 8*(15
0*a^2*b^4*c^2*d^3 - 175*a^3*b^3*c*d^4 + 63*a^4*b^2*d^5)*x^5 + 5*(9*b^6*c^5 + 15*
a*b^5*c^4*d - 150*a^2*b^4*c^3*d^2 + 750*a^3*b^3*c^2*d^3 - 875*a^4*b^2*c*d^4 + 31
5*a^5*b*d^5)*x^3 + 15*(5*a*b^5*c^5 - 5*a^2*b^4*c^4*d - 30*a^3*b^3*c^3*d^2 + 150*
a^4*b^2*c^2*d^3 - 175*a^5*b*c*d^4 + 63*a^6*d^5)*x)*sqrt(a*b))/((a^2*b^7*x^4 + 2*
a^3*b^6*x^2 + a^4*b^5)*sqrt(a*b))]

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Sympy [A]  time = 21.4133, size = 614, normalized size = 3.13 \[ \frac{\sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \log{\left (- \frac{a^{3} b^{5} \sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{63 a^{5} d^{5} - 175 a^{4} b c d^{4} + 150 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d - 3 b^{5} c^{5}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right ) \log{\left (\frac{a^{3} b^{5} \sqrt{- \frac{1}{a^{5} b^{11}}} \left (a d - b c\right )^{3} \left (63 a^{2} d^{2} + 14 a b c d + 3 b^{2} c^{2}\right )}{63 a^{5} d^{5} - 175 a^{4} b c d^{4} + 150 a^{3} b^{2} c^{2} d^{3} - 30 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d - 3 b^{5} c^{5}} + x \right )}}{16} + \frac{x^{3} \left (17 a^{5} b d^{5} - 65 a^{4} b^{2} c d^{4} + 90 a^{3} b^{3} c^{2} d^{3} - 50 a^{2} b^{4} c^{3} d^{2} + 5 a b^{5} c^{4} d + 3 b^{6} c^{5}\right ) + x \left (15 a^{6} d^{5} - 55 a^{5} b c d^{4} + 70 a^{4} b^{2} c^{2} d^{3} - 30 a^{3} b^{3} c^{3} d^{2} - 5 a^{2} b^{4} c^{4} d + 5 a b^{5} c^{5}\right )}{8 a^{4} b^{5} + 16 a^{3} b^{6} x^{2} + 8 a^{2} b^{7} x^{4}} + \frac{d^{5} x^{5}}{5 b^{3}} - \frac{x^{3} \left (3 a d^{5} - 5 b c d^{4}\right )}{3 b^{4}} + \frac{x \left (6 a^{2} d^{5} - 15 a b c d^{4} + 10 b^{2} c^{2} d^{3}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)*l
og(-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) - x**3*(3*a*d**5 - 5*b*c*d**4)/(
3*b**4) + x*(6*a**2*d**5 - 15*a*b*c*d**4 + 10*b**2*c**2*d**3)/b**5

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GIAC/XCAS [A]  time = 0.232694, size = 459, normalized size = 2.34 \[ \frac{{\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d + 30 \, a^{2} b^{3} c^{3} d^{2} - 150 \, a^{3} b^{2} c^{2} d^{3} + 175 \, a^{4} b c d^{4} - 63 \, a^{5} d^{5}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{5}} + \frac{3 \, b^{6} c^{5} x^{3} + 5 \, a b^{5} c^{4} d x^{3} - 50 \, a^{2} b^{4} c^{3} d^{2} x^{3} + 90 \, a^{3} b^{3} c^{2} d^{3} x^{3} - 65 \, a^{4} b^{2} c d^{4} x^{3} + 17 \, a^{5} b d^{5} x^{3} + 5 \, a b^{5} c^{5} x - 5 \, a^{2} b^{4} c^{4} d x - 30 \, a^{3} b^{3} c^{3} d^{2} x + 70 \, a^{4} b^{2} c^{2} d^{3} x - 55 \, a^{5} b c d^{4} x + 15 \, a^{6} d^{5} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{5}} + \frac{3 \, b^{12} d^{5} x^{5} + 25 \, b^{12} c d^{4} x^{3} - 15 \, a b^{11} d^{5} x^{3} + 150 \, b^{12} c^{2} d^{3} x - 225 \, a b^{11} c d^{4} x + 90 \, a^{2} b^{10} d^{5} x}{15 \, b^{15}} \]

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)*arctan(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 - 3
0*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