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

Optimal. Leaf size=217 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{c x}-\frac{5 c^2 \left (b^2 c^2-16 a d (3 a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{3/2}}-\frac{x \left (c+d x^2\right )^{5/2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{48 c d}-\frac{5 x \left (c+d x^2\right )^{3/2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{192 d}-\frac{5 c x \sqrt{c+d x^2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{128 d}+\frac{b^2 x \left (c+d x^2\right )^{7/2}}{8 d} \]

[Out]

(-5*c*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*x*Sqrt[c + d*x^2])/(128*d) - (5*(b^2*c^2
- 16*a*d*(b*c + 3*a*d))*x*(c + d*x^2)^(3/2))/(192*d) - ((b^2*c^2 - 16*a*d*(b*c +
3*a*d))*x*(c + d*x^2)^(5/2))/(48*c*d) - (a^2*(c + d*x^2)^(7/2))/(c*x) + (b^2*x*
(c + d*x^2)^(7/2))/(8*d) - (5*c^2*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*ArcTanh[(Sqrt
[d]*x)/Sqrt[c + d*x^2]])/(128*d^(3/2))

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Rubi [A]  time = 0.360751, antiderivative size = 214, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1}{48} x \left (c+d x^2\right )^{5/2} \left (\frac{48 a^2 d}{c}+16 a b-\frac{b^2 c}{d}\right )-\frac{a^2 \left (c+d x^2\right )^{7/2}}{c x}-\frac{5 c^2 \left (b^2 c^2-16 a d (3 a d+b c)\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 d^{3/2}}-\frac{5 x \left (c+d x^2\right )^{3/2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{192 d}-\frac{5 c x \sqrt{c+d x^2} \left (b^2 c^2-16 a d (3 a d+b c)\right )}{128 d}+\frac{b^2 x \left (c+d x^2\right )^{7/2}}{8 d} \]

Antiderivative was successfully verified.

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

[Out]

(-5*c*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*x*Sqrt[c + d*x^2])/(128*d) - (5*(b^2*c^2
- 16*a*d*(b*c + 3*a*d))*x*(c + d*x^2)^(3/2))/(192*d) + ((16*a*b - (b^2*c)/d + (4
8*a^2*d)/c)*x*(c + d*x^2)^(5/2))/48 - (a^2*(c + d*x^2)^(7/2))/(c*x) + (b^2*x*(c
+ d*x^2)^(7/2))/(8*d) - (5*c^2*(b^2*c^2 - 16*a*d*(b*c + 3*a*d))*ArcTanh[(Sqrt[d]
*x)/Sqrt[c + d*x^2]])/(128*d^(3/2))

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Rubi in Sympy [A]  time = 32.4377, size = 199, normalized size = 0.92 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{c x} + \frac{b^{2} x \left (c + d x^{2}\right )^{\frac{7}{2}}}{8 d} - \frac{5 c^{2} \left (- 16 a d \left (3 a d + b c\right ) + b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{128 d^{\frac{3}{2}}} - \frac{5 c x \sqrt{c + d x^{2}} \left (- 16 a d \left (3 a d + b c\right ) + b^{2} c^{2}\right )}{128 d} - \frac{5 x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (- 16 a d \left (3 a d + b c\right ) + b^{2} c^{2}\right )}{192 d} - \frac{x \left (c + d x^{2}\right )^{\frac{5}{2}} \left (- 16 a d \left (3 a d + b c\right ) + b^{2} c^{2}\right )}{48 c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*(c + d*x**2)**(7/2)/(c*x) + b**2*x*(c + d*x**2)**(7/2)/(8*d) - 5*c**2*(-16
*a*d*(3*a*d + b*c) + b**2*c**2)*atanh(sqrt(d)*x/sqrt(c + d*x**2))/(128*d**(3/2))
- 5*c*x*sqrt(c + d*x**2)*(-16*a*d*(3*a*d + b*c) + b**2*c**2)/(128*d) - 5*x*(c +
d*x**2)**(3/2)*(-16*a*d*(3*a*d + b*c) + b**2*c**2)/(192*d) - x*(c + d*x**2)**(5
/2)*(-16*a*d*(3*a*d + b*c) + b**2*c**2)/(48*c*d)

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Mathematica [A]  time = 0.201584, size = 174, normalized size = 0.8 \[ \sqrt{c+d x^2} \left (\frac{1}{192} x^3 \left (48 a^2 d^2+208 a b c d+59 b^2 c^2\right )+\frac{c x \left (144 a^2 d^2+176 a b c d+5 b^2 c^2\right )}{128 d}-\frac{a^2 c^2}{x}+\frac{1}{48} b d x^5 (16 a d+17 b c)+\frac{1}{8} b^2 d^2 x^7\right )-\frac{5 c^2 \left (-48 a^2 d^2-16 a b c d+b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{128 d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[c + d*x^2]*(-((a^2*c^2)/x) + (c*(5*b^2*c^2 + 176*a*b*c*d + 144*a^2*d^2)*x)/
(128*d) + ((59*b^2*c^2 + 208*a*b*c*d + 48*a^2*d^2)*x^3)/192 + (b*d*(17*b*c + 16*
a*d)*x^5)/48 + (b^2*d^2*x^7)/8) - (5*c^2*(b^2*c^2 - 16*a*b*c*d - 48*a^2*d^2)*Log
[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/(128*d^(3/2))

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Maple [A]  time = 0.017, size = 278, normalized size = 1.3 \[{\frac{{b}^{2}x}{8\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{2}cx}{48\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}{c}^{2}x}{192\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x{b}^{2}{c}^{3}}{128\,d}\sqrt{d{x}^{2}+c}}-{\frac{5\,{b}^{2}{c}^{4}}{128}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){d}^{-{\frac{3}{2}}}}-{\frac{{a}^{2}}{cx} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}dx}{c} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}dx}{4} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{a}^{2}cdx}{8}\sqrt{d{x}^{2}+c}}+{\frac{15\,{a}^{2}{c}^{2}}{8}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }+{\frac{abx}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,abcx}{12} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{5\,ab{c}^{2}x}{8}\sqrt{d{x}^{2}+c}}+{\frac{5\,ab{c}^{3}}{8}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ){\frac{1}{\sqrt{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*x*(d*x^2+c)^(7/2)/d-1/48*b^2*c/d*x*(d*x^2+c)^(5/2)-5/192*b^2*c^2/d*x*(d*
x^2+c)^(3/2)-5/128*b^2*c^3/d*x*(d*x^2+c)^(1/2)-5/128*b^2*c^4/d^(3/2)*ln(x*d^(1/2
)+(d*x^2+c)^(1/2))-a^2*(d*x^2+c)^(7/2)/c/x+a^2*d/c*x*(d*x^2+c)^(5/2)+5/4*a^2*d*x
*(d*x^2+c)^(3/2)+15/8*a^2*d*c*x*(d*x^2+c)^(1/2)+15/8*a^2*d^(1/2)*c^2*ln(x*d^(1/2
)+(d*x^2+c)^(1/2))+1/3*a*b*x*(d*x^2+c)^(5/2)+5/12*a*b*c*x*(d*x^2+c)^(3/2)+5/8*a*
b*c^2*x*(d*x^2+c)^(1/2)+5/8*a*b*c^3/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.362638, size = 1, normalized size = 0. \[ \left [-\frac{15 \,{\left (b^{2} c^{4} - 16 \, a b c^{3} d - 48 \, a^{2} c^{2} d^{2}\right )} x \log \left (-2 \, \sqrt{d x^{2} + c} d x -{\left (2 \, d x^{2} + c\right )} \sqrt{d}\right ) - 2 \,{\left (48 \, b^{2} d^{3} x^{8} + 8 \,{\left (17 \, b^{2} c d^{2} + 16 \, a b d^{3}\right )} x^{6} - 384 \, a^{2} c^{2} d + 2 \,{\left (59 \, b^{2} c^{2} d + 208 \, a b c d^{2} + 48 \, a^{2} d^{3}\right )} x^{4} + 3 \,{\left (5 \, b^{2} c^{3} + 176 \, a b c^{2} d + 144 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{d}}{768 \, d^{\frac{3}{2}} x}, -\frac{15 \,{\left (b^{2} c^{4} - 16 \, a b c^{3} d - 48 \, a^{2} c^{2} d^{2}\right )} x \arctan \left (\frac{\sqrt{-d} x}{\sqrt{d x^{2} + c}}\right ) -{\left (48 \, b^{2} d^{3} x^{8} + 8 \,{\left (17 \, b^{2} c d^{2} + 16 \, a b d^{3}\right )} x^{6} - 384 \, a^{2} c^{2} d + 2 \,{\left (59 \, b^{2} c^{2} d + 208 \, a b c d^{2} + 48 \, a^{2} d^{3}\right )} x^{4} + 3 \,{\left (5 \, b^{2} c^{3} + 176 \, a b c^{2} d + 144 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-d}}{384 \, \sqrt{-d} d x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/768*(15*(b^2*c^4 - 16*a*b*c^3*d - 48*a^2*c^2*d^2)*x*log(-2*sqrt(d*x^2 + c)*d
*x - (2*d*x^2 + c)*sqrt(d)) - 2*(48*b^2*d^3*x^8 + 8*(17*b^2*c*d^2 + 16*a*b*d^3)*
x^6 - 384*a^2*c^2*d + 2*(59*b^2*c^2*d + 208*a*b*c*d^2 + 48*a^2*d^3)*x^4 + 3*(5*b
^2*c^3 + 176*a*b*c^2*d + 144*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(d))/(d^(3/2)*x
), -1/384*(15*(b^2*c^4 - 16*a*b*c^3*d - 48*a^2*c^2*d^2)*x*arctan(sqrt(-d)*x/sqrt
(d*x^2 + c)) - (48*b^2*d^3*x^8 + 8*(17*b^2*c*d^2 + 16*a*b*d^3)*x^6 - 384*a^2*c^2
*d + 2*(59*b^2*c^2*d + 208*a*b*c*d^2 + 48*a^2*d^3)*x^4 + 3*(5*b^2*c^3 + 176*a*b*
c^2*d + 144*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c)*sqrt(-d))/(sqrt(-d)*d*x)]

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Sympy [A]  time = 117.643, size = 496, normalized size = 2.29 \[ - \frac{a^{2} c^{\frac{5}{2}}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + a^{2} c^{\frac{3}{2}} d x \sqrt{1 + \frac{d x^{2}}{c}} - \frac{7 a^{2} c^{\frac{3}{2}} d x}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 a^{2} \sqrt{c} d^{2} x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{15 a^{2} c^{2} \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8} + \frac{a^{2} d^{3} x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + a b c^{\frac{5}{2}} x \sqrt{1 + \frac{d x^{2}}{c}} + \frac{3 a b c^{\frac{5}{2}} x}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{35 a b c^{\frac{3}{2}} d x^{3}}{24 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{17 a b \sqrt{c} d^{2} x^{5}}{12 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 a b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{d}} + \frac{a b d^{3} x^{7}}{3 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{5 b^{2} c^{\frac{7}{2}} x}{128 d \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{133 b^{2} c^{\frac{5}{2}} x^{3}}{384 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{127 b^{2} c^{\frac{3}{2}} d x^{5}}{192 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{23 b^{2} \sqrt{c} d^{2} x^{7}}{48 \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{5 b^{2} c^{4} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{128 d^{\frac{3}{2}}} + \frac{b^{2} d^{3} x^{9}}{8 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*c**(5/2)/(x*sqrt(1 + d*x**2/c)) + a**2*c**(3/2)*d*x*sqrt(1 + d*x**2/c) - 7
*a**2*c**(3/2)*d*x/(8*sqrt(1 + d*x**2/c)) + 3*a**2*sqrt(c)*d**2*x**3/(8*sqrt(1 +
d*x**2/c)) + 15*a**2*c**2*sqrt(d)*asinh(sqrt(d)*x/sqrt(c))/8 + a**2*d**3*x**5/(
4*sqrt(c)*sqrt(1 + d*x**2/c)) + a*b*c**(5/2)*x*sqrt(1 + d*x**2/c) + 3*a*b*c**(5/
2)*x/(8*sqrt(1 + d*x**2/c)) + 35*a*b*c**(3/2)*d*x**3/(24*sqrt(1 + d*x**2/c)) + 1
7*a*b*sqrt(c)*d**2*x**5/(12*sqrt(1 + d*x**2/c)) + 5*a*b*c**3*asinh(sqrt(d)*x/sqr
t(c))/(8*sqrt(d)) + a*b*d**3*x**7/(3*sqrt(c)*sqrt(1 + d*x**2/c)) + 5*b**2*c**(7/
2)*x/(128*d*sqrt(1 + d*x**2/c)) + 133*b**2*c**(5/2)*x**3/(384*sqrt(1 + d*x**2/c)
) + 127*b**2*c**(3/2)*d*x**5/(192*sqrt(1 + d*x**2/c)) + 23*b**2*sqrt(c)*d**2*x**
7/(48*sqrt(1 + d*x**2/c)) - 5*b**2*c**4*asinh(sqrt(d)*x/sqrt(c))/(128*d**(3/2))
+ b**2*d**3*x**9/(8*sqrt(c)*sqrt(1 + d*x**2/c))

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GIAC/XCAS [A]  time = 0.248285, size = 296, normalized size = 1.36 \[ \frac{2 \, a^{2} c^{3} \sqrt{d}}{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c} + \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (6 \, b^{2} d^{2} x^{2} + \frac{17 \, b^{2} c d^{7} + 16 \, a b d^{8}}{d^{6}}\right )} x^{2} + \frac{59 \, b^{2} c^{2} d^{6} + 208 \, a b c d^{7} + 48 \, a^{2} d^{8}}{d^{6}}\right )} x^{2} + \frac{3 \,{\left (5 \, b^{2} c^{3} d^{5} + 176 \, a b c^{2} d^{6} + 144 \, a^{2} c d^{7}\right )}}{d^{6}}\right )} \sqrt{d x^{2} + c} x + \frac{5 \,{\left (b^{2} c^{4} \sqrt{d} - 16 \, a b c^{3} d^{\frac{3}{2}} - 48 \, a^{2} c^{2} d^{\frac{5}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{256 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a^2*c^3*sqrt(d)/((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c) + 1/384*(2*(4*(6*b^2*d^2
*x^2 + (17*b^2*c*d^7 + 16*a*b*d^8)/d^6)*x^2 + (59*b^2*c^2*d^6 + 208*a*b*c*d^7 +
48*a^2*d^8)/d^6)*x^2 + 3*(5*b^2*c^3*d^5 + 176*a*b*c^2*d^6 + 144*a^2*c*d^7)/d^6)*
sqrt(d*x^2 + c)*x + 5/256*(b^2*c^4*sqrt(d) - 16*a*b*c^3*d^(3/2) - 48*a^2*c^2*d^(
5/2))*ln((sqrt(d)*x - sqrt(d*x^2 + c))^2)/d^2