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

Optimal. Leaf size=162 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}-\frac{1}{2} a c^{3/2} (5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+\frac{a \left (c+d x^2\right )^{5/2} (5 a d+4 b c)}{10 c}+\frac{1}{6} a \left (c+d x^2\right )^{3/2} (5 a d+4 b c)+\frac{1}{2} a c \sqrt{c+d x^2} (5 a d+4 b c)+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d} \]

[Out]

(a*c*(4*b*c + 5*a*d)*Sqrt[c + d*x^2])/2 + (a*(4*b*c + 5*a*d)*(c + d*x^2)^(3/2))/
6 + (a*(4*b*c + 5*a*d)*(c + d*x^2)^(5/2))/(10*c) + (b^2*(c + d*x^2)^(7/2))/(7*d)
- (a^2*(c + d*x^2)^(7/2))/(2*c*x^2) - (a*c^(3/2)*(4*b*c + 5*a*d)*ArcTanh[Sqrt[c
+ d*x^2]/Sqrt[c]])/2

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Rubi [A]  time = 0.363152, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{2 c x^2}-\frac{1}{2} a c^{3/2} (5 a d+4 b c) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )+\frac{a \left (c+d x^2\right )^{5/2} (5 a d+4 b c)}{10 c}+\frac{1}{6} a \left (c+d x^2\right )^{3/2} (5 a d+4 b c)+\frac{1}{2} a c \sqrt{c+d x^2} (5 a d+4 b c)+\frac{b^2 \left (c+d x^2\right )^{7/2}}{7 d} \]

Antiderivative was successfully verified.

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

[Out]

(a*c*(4*b*c + 5*a*d)*Sqrt[c + d*x^2])/2 + (a*(4*b*c + 5*a*d)*(c + d*x^2)^(3/2))/
6 + (a*(4*b*c + 5*a*d)*(c + d*x^2)^(5/2))/(10*c) + (b^2*(c + d*x^2)^(7/2))/(7*d)
- (a^2*(c + d*x^2)^(7/2))/(2*c*x^2) - (a*c^(3/2)*(4*b*c + 5*a*d)*ArcTanh[Sqrt[c
+ d*x^2]/Sqrt[c]])/2

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Rubi in Sympy [A]  time = 31.966, size = 146, normalized size = 0.9 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{2 c x^{2}} - \frac{a c^{\frac{3}{2}} \left (5 a d + 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{2} + \frac{a c \sqrt{c + d x^{2}} \left (5 a d + 4 b c\right )}{2} + \frac{a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (5 a d + 4 b c\right )}{6} + \frac{a \left (c + d x^{2}\right )^{\frac{5}{2}} \left (5 a d + 4 b c\right )}{10 c} + \frac{b^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{7 d} \]

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**3,x)

[Out]

-a**2*(c + d*x**2)**(7/2)/(2*c*x**2) - a*c**(3/2)*(5*a*d + 4*b*c)*atanh(sqrt(c +
d*x**2)/sqrt(c))/2 + a*c*sqrt(c + d*x**2)*(5*a*d + 4*b*c)/2 + a*(c + d*x**2)**(
3/2)*(5*a*d + 4*b*c)/6 + a*(c + d*x**2)**(5/2)*(5*a*d + 4*b*c)/(10*c) + b**2*(c
+ d*x**2)**(7/2)/(7*d)

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Mathematica [A]  time = 0.375689, size = 156, normalized size = 0.96 \[ \frac{\sqrt{c+d x^2} \left (35 a^2 d \left (-3 c^2+14 c d x^2+2 d^2 x^4\right )+28 a b d x^2 \left (23 c^2+11 c d x^2+3 d^2 x^4\right )+30 b^2 x^2 \left (c+d x^2\right )^3\right )}{210 d x^2}-\frac{1}{2} a c^{3/2} (5 a d+4 b c) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+\frac{1}{2} a c^{3/2} \log (x) (5 a d+4 b c) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x^2]*(30*b^2*x^2*(c + d*x^2)^3 + 35*a^2*d*(-3*c^2 + 14*c*d*x^2 + 2*d
^2*x^4) + 28*a*b*d*x^2*(23*c^2 + 11*c*d*x^2 + 3*d^2*x^4)))/(210*d*x^2) + (a*c^(3
/2)*(4*b*c + 5*a*d)*Log[x])/2 - (a*c^(3/2)*(4*b*c + 5*a*d)*Log[c + Sqrt[c]*Sqrt[
c + d*x^2]])/2

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Maple [A]  time = 0.018, size = 193, normalized size = 1.2 \[{\frac{{b}^{2}}{7\,d} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{a}^{2}d}{2\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{a}^{2}d}{6} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{a}^{2}d}{2}{c}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) }+{\frac{5\,{a}^{2}cd}{2}\sqrt{d{x}^{2}+c}}+{\frac{2\,ab}{5} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,abc}{3} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-2\,ab{c}^{5/2}\ln \left ({\frac{2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c}}{x}} \right ) +2\,ab\sqrt{d{x}^{2}+c}{c}^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^2*(d*x^2+c)^(7/2)/d-1/2*a^2*(d*x^2+c)^(7/2)/c/x^2+1/2*a^2*d/c*(d*x^2+c)^(5
/2)+5/6*a^2*d*(d*x^2+c)^(3/2)-5/2*a^2*d*c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2
))/x)+5/2*a^2*d*c*(d*x^2+c)^(1/2)+2/5*a*b*(d*x^2+c)^(5/2)+2/3*a*b*c*(d*x^2+c)^(3
/2)-2*a*b*c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+2*a*b*(d*x^2+c)^(1/2)*c^
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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258755, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (4 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} \sqrt{c} x^{2} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right ) + 2 \,{\left (30 \, b^{2} d^{3} x^{8} + 6 \,{\left (15 \, b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{6} - 105 \, a^{2} c^{2} d + 2 \,{\left (45 \, b^{2} c^{2} d + 154 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{4} + 2 \,{\left (15 \, b^{2} c^{3} + 322 \, a b c^{2} d + 245 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{420 \, d x^{2}}, -\frac{105 \,{\left (4 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} \sqrt{-c} x^{2} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) -{\left (30 \, b^{2} d^{3} x^{8} + 6 \,{\left (15 \, b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{6} - 105 \, a^{2} c^{2} d + 2 \,{\left (45 \, b^{2} c^{2} d + 154 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{4} + 2 \,{\left (15 \, b^{2} c^{3} + 322 \, a b c^{2} d + 245 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{210 \, d x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x^3,x, algorithm="fricas")

[Out]

[1/420*(105*(4*a*b*c^2*d + 5*a^2*c*d^2)*sqrt(c)*x^2*log(-(d*x^2 - 2*sqrt(d*x^2 +
c)*sqrt(c) + 2*c)/x^2) + 2*(30*b^2*d^3*x^8 + 6*(15*b^2*c*d^2 + 14*a*b*d^3)*x^6
- 105*a^2*c^2*d + 2*(45*b^2*c^2*d + 154*a*b*c*d^2 + 35*a^2*d^3)*x^4 + 2*(15*b^2*
c^3 + 322*a*b*c^2*d + 245*a^2*c*d^2)*x^2)*sqrt(d*x^2 + c))/(d*x^2), -1/210*(105*
(4*a*b*c^2*d + 5*a^2*c*d^2)*sqrt(-c)*x^2*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) -
(30*b^2*d^3*x^8 + 6*(15*b^2*c*d^2 + 14*a*b*d^3)*x^6 - 105*a^2*c^2*d + 2*(45*b^2*
c^2*d + 154*a*b*c*d^2 + 35*a^2*d^3)*x^4 + 2*(15*b^2*c^3 + 322*a*b*c^2*d + 245*a^
2*c*d^2)*x^2)*sqrt(d*x^2 + c))/(d*x^2)]

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Sympy [A]  time = 71.7866, size = 518, normalized size = 3.2 \[ - \frac{5 a^{2} c^{\frac{3}{2}} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2} - \frac{a^{2} c^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} + \frac{2 a^{2} c^{2} \sqrt{d}}{x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 a^{2} c d^{\frac{3}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + a^{2} d^{2} \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) - 2 a b c^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )} + \frac{2 a b c^{3}}{\sqrt{d} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{2 a b c^{2} \sqrt{d} x}{\sqrt{\frac{c}{d x^{2}} + 1}} + 4 a b c d \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + 2 a b d^{2} \left (\begin{cases} - \frac{2 c^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{c x^{2} \sqrt{c + d x^{2}}}{15 d} + \frac{x^{4} \sqrt{c + d x^{2}}}{5} & \text{for}\: d \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + b^{2} c^{2} \left (\begin{cases} \frac{\sqrt{c} x^{2}}{2} & \text{for}\: d = 0 \\\frac{\left (c + d x^{2}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + 2 b^{2} c d \left (\begin{cases} - \frac{2 c^{2} \sqrt{c + d x^{2}}}{15 d^{2}} + \frac{c x^{2} \sqrt{c + d x^{2}}}{15 d} + \frac{x^{4} \sqrt{c + d x^{2}}}{5} & \text{for}\: d \neq 0 \\\frac{\sqrt{c} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + b^{2} d^{2} \left (\begin{cases} \frac{8 c^{3} \sqrt{c + d x^{2}}}{105 d^{3}} - \frac{4 c^{2} x^{2} \sqrt{c + d x^{2}}}{105 d^{2}} + \frac{c x^{4} \sqrt{c + d x^{2}}}{35 d} + \frac{x^{6} \sqrt{c + d x^{2}}}{7} & \text{for}\: d \neq 0 \\\frac{\sqrt{c} x^{6}}{6} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**2*c**(3/2)*d*asinh(sqrt(c)/(sqrt(d)*x))/2 - a**2*c**2*sqrt(d)*sqrt(c/(d*x*
*2) + 1)/(2*x) + 2*a**2*c**2*sqrt(d)/(x*sqrt(c/(d*x**2) + 1)) + 2*a**2*c*d**(3/2
)*x/sqrt(c/(d*x**2) + 1) + a**2*d**2*Piecewise((sqrt(c)*x**2/2, Eq(d, 0)), ((c +
d*x**2)**(3/2)/(3*d), True)) - 2*a*b*c**(5/2)*asinh(sqrt(c)/(sqrt(d)*x)) + 2*a*
b*c**3/(sqrt(d)*x*sqrt(c/(d*x**2) + 1)) + 2*a*b*c**2*sqrt(d)*x/sqrt(c/(d*x**2) +
1) + 4*a*b*c*d*Piecewise((sqrt(c)*x**2/2, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d)
, True)) + 2*a*b*d**2*Piecewise((-2*c**2*sqrt(c + d*x**2)/(15*d**2) + c*x**2*sqr
t(c + d*x**2)/(15*d) + x**4*sqrt(c + d*x**2)/5, Ne(d, 0)), (sqrt(c)*x**4/4, True
)) + b**2*c**2*Piecewise((sqrt(c)*x**2/2, Eq(d, 0)), ((c + d*x**2)**(3/2)/(3*d),
True)) + 2*b**2*c*d*Piecewise((-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)
) + b**2*d**2*Piecewise((8*c**3*sqrt(c + d*x**2)/(105*d**3) - 4*c**2*x**2*sqrt(c
+ d*x**2)/(105*d**2) + c*x**4*sqrt(c + d*x**2)/(35*d) + x**6*sqrt(c + d*x**2)/7
, Ne(d, 0)), (sqrt(c)*x**6/6, True))

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GIAC/XCAS [A]  time = 0.245608, size = 223, normalized size = 1.38 \[ \frac{30 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} b^{2} + 84 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b d + 140 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c d + 420 \, \sqrt{d x^{2} + c} a b c^{2} d + 70 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} d^{2} + 420 \, \sqrt{d x^{2} + c} a^{2} c d^{2} - \frac{105 \, \sqrt{d x^{2} + c} a^{2} c^{2} d}{x^{2}} + \frac{105 \,{\left (4 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c}}}{210 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x^3,x, algorithm="giac")

[Out]

1/210*(30*(d*x^2 + c)^(7/2)*b^2 + 84*(d*x^2 + c)^(5/2)*a*b*d + 140*(d*x^2 + c)^(
3/2)*a*b*c*d + 420*sqrt(d*x^2 + c)*a*b*c^2*d + 70*(d*x^2 + c)^(3/2)*a^2*d^2 + 42
0*sqrt(d*x^2 + c)*a^2*c*d^2 - 105*sqrt(d*x^2 + c)*a^2*c^2*d/x^2 + 105*(4*a*b*c^3
*d + 5*a^2*c^2*d^2)*arctan(sqrt(d*x^2 + c)/sqrt(-c))/sqrt(-c))/d