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

Optimal. Leaf size=136 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{a \left (c+d x^2\right )^{3/2} (3 a d+4 b c)}{6 c}+\frac{1}{2} a \sqrt{c+d x^2} (3 a d+4 b c)-\frac{1}{2} a \sqrt{c} (3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d} \]

[Out]

(a*(4*b*c + 3*a*d)*Sqrt[c + d*x^2])/2 + (a*(4*b*c + 3*a*d)*(c + d*x^2)^(3/2))/(6
*c) + (b^2*(c + d*x^2)^(5/2))/(5*d) - (a^2*(c + d*x^2)^(5/2))/(2*c*x^2) - (a*Sqr
t[c]*(4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/2

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Rubi [A]  time = 0.314524, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{2 c x^2}+\frac{a \left (c+d x^2\right )^{3/2} (3 a d+4 b c)}{6 c}+\frac{1}{2} a \sqrt{c+d x^2} (3 a d+4 b c)-\frac{1}{2} a \sqrt{c} (3 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+\frac{b^2 \left (c+d x^2\right )^{5/2}}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

(a*(4*b*c + 3*a*d)*Sqrt[c + d*x^2])/2 + (a*(4*b*c + 3*a*d)*(c + d*x^2)^(3/2))/(6
*c) + (b^2*(c + d*x^2)^(5/2))/(5*d) - (a^2*(c + d*x^2)^(5/2))/(2*c*x^2) - (a*Sqr
t[c]*(4*b*c + 3*a*d)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/2

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Rubi in Sympy [A]  time = 27.9572, size = 121, normalized size = 0.89 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{2 c x^{2}} - \frac{a \sqrt{c} \left (3 a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2} + \frac{a \sqrt{c + d x^{2}} \left (3 a d + 4 b c\right )}{2} + \frac{a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (3 a d + 4 b c\right )}{6 c} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*(c + d*x**2)**(5/2)/(2*c*x**2) - a*sqrt(c)*(3*a*d + 4*b*c)*atanh(sqrt(c +
d*x**2)/sqrt(c))/2 + a*sqrt(c + d*x**2)*(3*a*d + 4*b*c)/2 + a*(c + d*x**2)**(3/2
)*(3*a*d + 4*b*c)/(6*c) + b**2*(c + d*x**2)**(5/2)/(5*d)

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Mathematica [A]  time = 0.270848, size = 128, normalized size = 0.94 \[ \frac{1}{30} \left (\frac{\sqrt{c+d x^2} \left (-15 a^2 d \left (c-2 d x^2\right )+20 a b d x^2 \left (4 c+d x^2\right )+6 b^2 x^2 \left (c+d x^2\right )^2\right )}{d x^2}-15 a \sqrt{c} (3 a d+4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+15 a \sqrt{c} \log (x) (3 a d+4 b c)\right ) \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[c + d*x^2]*(-15*a^2*d*(c - 2*d*x^2) + 6*b^2*x^2*(c + d*x^2)^2 + 20*a*b*d*
x^2*(4*c + d*x^2)))/(d*x^2) + 15*a*Sqrt[c]*(4*b*c + 3*a*d)*Log[x] - 15*a*Sqrt[c]
*(4*b*c + 3*a*d)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/30

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Maple [A]  time = 0.016, size = 161, normalized size = 1.2 \[{\frac{{b}^{2}}{5\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}d}{2\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{a}^{2}d}{2}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) }+{\frac{3\,{a}^{2}d}{2}\sqrt{d{x}^{2}+c}}+{\frac{2\,ab}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-2\,ab\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ){c}^{3/2}+2\,ab\sqrt{d{x}^{2}+c}c \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*(d*x^2+c)^(5/2)/d-1/2*a^2*(d*x^2+c)^(5/2)/c/x^2+1/2*a^2*d/c*(d*x^2+c)^(3
/2)-3/2*a^2*d*c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+3/2*a^2*d*(d*x^2+c)^
(1/2)+2/3*a*b*(d*x^2+c)^(3/2)-2*a*b*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)*c^(3/2
)+2*a*b*(d*x^2+c)^(1/2)*c

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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)^(3/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23984, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (4 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{c} x^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (6 \, b^{2} d^{2} x^{6} + 4 \,{\left (3 \, b^{2} c d + 5 \, a b d^{2}\right )} x^{4} - 15 \, a^{2} c d + 2 \,{\left (3 \, b^{2} c^{2} + 40 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{60 \, d x^{2}}, -\frac{15 \,{\left (4 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt{-c} x^{2} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (6 \, b^{2} d^{2} x^{6} + 4 \,{\left (3 \, b^{2} c d + 5 \, a b d^{2}\right )} x^{4} - 15 \, a^{2} c d + 2 \,{\left (3 \, b^{2} c^{2} + 40 \, a b c d + 15 \, a^{2} d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{30 \, d x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/60*(15*(4*a*b*c*d + 3*a^2*d^2)*sqrt(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sq
rt(c) + 2*c)/x^2) + 2*(6*b^2*d^2*x^6 + 4*(3*b^2*c*d + 5*a*b*d^2)*x^4 - 15*a^2*c*
d + 2*(3*b^2*c^2 + 40*a*b*c*d + 15*a^2*d^2)*x^2)*sqrt(d*x^2 + c))/(d*x^2), -1/30
*(15*(4*a*b*c*d + 3*a^2*d^2)*sqrt(-c)*x^2*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) -
(6*b^2*d^2*x^6 + 4*(3*b^2*c*d + 5*a*b*d^2)*x^4 - 15*a^2*c*d + 2*(3*b^2*c^2 + 40
*a*b*c*d + 15*a^2*d^2)*x^2)*sqrt(d*x^2 + c))/(d*x^2)]

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Sympy [A]  time = 47.0053, size = 303, normalized size = 2.23 \[ - \frac{3 a^{2} \sqrt{c} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2} - \frac{a^{2} c \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} + \frac{a^{2} c \sqrt{d}}{x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{\frac{3}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} - 2 a b c^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} + \frac{2 a b c^{2}}{\sqrt{d} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 a b c \sqrt{d} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + 2 a b d \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + b^{2} c \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + b^{2} d \left (\begin{cases} - \frac{2 c^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{c x^{2} \sqrt{c + d x^{2}}}{15 d} + \frac{x^{4} \sqrt{c + d x^{2}}}{5} & \text{for}\: d \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**2*sqrt(c)*d*asinh(sqrt(c)/(sqrt(d)*x))/2 - a**2*c*sqrt(d)*sqrt(c/(d*x**2)
+ 1)/(2*x) + a**2*c*sqrt(d)/(x*sqrt(c/(d*x**2) + 1)) + a**2*d**(3/2)*x/sqrt(c/(d
*x**2) + 1) - 2*a*b*c**(3/2)*asinh(sqrt(c)/(sqrt(d)*x)) + 2*a*b*c**2/(sqrt(d)*x*
sqrt(c/(d*x**2) + 1)) + 2*a*b*c*sqrt(d)*x/sqrt(c/(d*x**2) + 1) + 2*a*b*d*Piecewi
se((sqrt(c)*x**2/2, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d), True)) + b**2*c*Piece
wise((sqrt(c)*x**2/2, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d), True)) + b**2*d*Pie
cewise((-2*c**2*sqrt(c + d*x**2)/(15*d**2) + c*x**2*sqrt(c + d*x**2)/(15*d) + x*
*4*sqrt(c + d*x**2)/5, Ne(d, 0)), (sqrt(c)*x**4/4, True))

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GIAC/XCAS [A]  time = 0.232335, size = 170, normalized size = 1.25 \[ \frac{6 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} + 20 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b d + 60 \, \sqrt{d x^{2} + c} a b c d + 30 \, \sqrt{d x^{2} + c} a^{2} d^{2} - \frac{15 \, \sqrt{d x^{2} + c} a^{2} c d}{x^{2}} + \frac{15 \,{\left (4 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}}}{30 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(6*(d*x^2 + c)^(5/2)*b^2 + 20*(d*x^2 + c)^(3/2)*a*b*d + 60*sqrt(d*x^2 + c)*
a*b*c*d + 30*sqrt(d*x^2 + c)*a^2*d^2 - 15*sqrt(d*x^2 + c)*a^2*c*d/x^2 + 15*(4*a*
b*c^2*d + 3*a^2*c*d^2)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c))/d