3.604 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^7} \, dx\)

Optimal. Leaf size=149 \[ -\frac{\sqrt{c+d x^2} \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{16 c^2 x^2}-\frac{d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{5/2}}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}-\frac{a \left (c+d x^2\right )^{3/2} (4 b c-a d)}{8 c^2 x^4} \]

[Out]

-((8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Sqrt[c + d*x^2])/(16*c^2*x^2) - (a^2*(c + d*
x^2)^(3/2))/(6*c*x^6) - (a*(4*b*c - a*d)*(c + d*x^2)^(3/2))/(8*c^2*x^4) - (d*(8*
b^2*c^2 - 4*a*b*c*d + a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(5/2))

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Rubi [A]  time = 0.382887, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{c+d x^2} \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{16 c^2 x^2}-\frac{d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{5/2}}-\frac{a^2 \left (c+d x^2\right )^{3/2}}{6 c x^6}-\frac{a \left (c+d x^2\right )^{3/2} (4 b c-a d)}{8 c^2 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^7,x]

[Out]

-((8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Sqrt[c + d*x^2])/(16*c^2*x^2) - (a^2*(c + d*
x^2)^(3/2))/(6*c*x^6) - (a*(4*b*c - a*d)*(c + d*x^2)^(3/2))/(8*c^2*x^4) - (d*(8*
b^2*c^2 - 4*a*b*c*d + a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(5/2))

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Rubi in Sympy [A]  time = 29.117, size = 133, normalized size = 0.89 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{6 c x^{6}} + \frac{a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - 4 b c\right )}{8 c^{2} x^{4}} - \frac{\sqrt{c + d x^{2}} \left (a d \left (a d - 4 b c\right ) + 8 b^{2} c^{2}\right )}{16 c^{2} x^{2}} - \frac{d \left (a d \left (a d - 4 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{16 c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**7,x)

[Out]

-a**2*(c + d*x**2)**(3/2)/(6*c*x**6) + a*(c + d*x**2)**(3/2)*(a*d - 4*b*c)/(8*c*
*2*x**4) - sqrt(c + d*x**2)*(a*d*(a*d - 4*b*c) + 8*b**2*c**2)/(16*c**2*x**2) - d
*(a*d*(a*d - 4*b*c) + 8*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(16*c**(5/2))

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Mathematica [A]  time = 0.206289, size = 161, normalized size = 1.08 \[ \sqrt{c+d x^2} \left (\frac{a^2 d^2-4 a b c d-8 b^2 c^2}{16 c^2 x^2}-\frac{a^2}{6 x^6}-\frac{a (a d+12 b c)}{24 c x^4}\right )-\frac{d \left (a^2 d^2-4 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{16 c^{5/2}}+\frac{d \log (x) \left (a^2 d^2-4 a b c d+8 b^2 c^2\right )}{16 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^7,x]

[Out]

(-a^2/(6*x^6) - (a*(12*b*c + a*d))/(24*c*x^4) + (-8*b^2*c^2 - 4*a*b*c*d + a^2*d^
2)/(16*c^2*x^2))*Sqrt[c + d*x^2] + (d*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[x])/
(16*c^(5/2)) - (d*(8*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*Log[c + Sqrt[c]*Sqrt[c + d*x
^2]])/(16*c^(5/2))

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Maple [B]  time = 0.018, size = 281, normalized size = 1.9 \[ -{\frac{{a}^{2}}{6\,c{x}^{6}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}d}{8\,{c}^{2}{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{2}}{16\,{c}^{3}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}+{\frac{{a}^{2}{d}^{3}}{16\,{c}^{3}}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{{b}^{2}d}{2\,c}\sqrt{d{x}^{2}+c}}-{\frac{ab}{2\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{abd}{4\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{ab{d}^{2}}{4}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{ab{d}^{2}}{4\,{c}^{2}}\sqrt{d{x}^{2}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^7,x)

[Out]

-1/6*a^2*(d*x^2+c)^(3/2)/c/x^6+1/8*a^2*d/c^2/x^4*(d*x^2+c)^(3/2)-1/16*a^2*d^2/c^
3/x^2*(d*x^2+c)^(3/2)-1/16*a^2*d^3/c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)
+1/16*a^2*d^3/c^3*(d*x^2+c)^(1/2)-1/2*b^2/c/x^2*(d*x^2+c)^(3/2)-1/2*b^2*d/c^(1/2
)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+1/2*b^2*d/c*(d*x^2+c)^(1/2)-1/2*a*b/c/x^
4*(d*x^2+c)^(3/2)+1/4*a*b*d/c^2/x^2*(d*x^2+c)^(3/2)+1/4*a*b*d^2/c^(3/2)*ln((2*c+
2*c^(1/2)*(d*x^2+c)^(1/2))/x)-1/4*a*b*d^2/c^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*sqrt(d*x^2 + c)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28499, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (8 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} - 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (8 \, b^{2} c^{2} + 4 \, a b c d - a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 2 \,{\left (12 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c}}{96 \, c^{\frac{5}{2}} x^{6}}, -\frac{3 \,{\left (8 \, b^{2} c^{2} d - 4 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) +{\left (3 \,{\left (8 \, b^{2} c^{2} + 4 \, a b c d - a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 2 \,{\left (12 \, a b c^{2} + a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{48 \, \sqrt{-c} c^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^7,x, algorithm="fricas")

[Out]

[1/96*(3*(8*b^2*c^2*d - 4*a*b*c*d^2 + a^2*d^3)*x^6*log(-((d*x^2 + 2*c)*sqrt(c) -
2*sqrt(d*x^2 + c)*c)/x^2) - 2*(3*(8*b^2*c^2 + 4*a*b*c*d - a^2*d^2)*x^4 + 8*a^2*
c^2 + 2*(12*a*b*c^2 + a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(c))/(c^(5/2)*x^6), -1/4
8*(3*(8*b^2*c^2*d - 4*a*b*c*d^2 + a^2*d^3)*x^6*arctan(sqrt(-c)/sqrt(d*x^2 + c))
+ (3*(8*b^2*c^2 + 4*a*b*c*d - a^2*d^2)*x^4 + 8*a^2*c^2 + 2*(12*a*b*c^2 + a^2*c*d
)*x^2)*sqrt(d*x^2 + c)*sqrt(-c))/(sqrt(-c)*c^2*x^6)]

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Sympy [A]  time = 96.0702, size = 291, normalized size = 1.95 \[ - \frac{a^{2} c}{6 \sqrt{d} x^{7} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{5 a^{2} \sqrt{d}}{24 x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{\frac{3}{2}}}{48 c x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{\frac{5}{2}}}{16 c^{2} x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a^{2} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{16 c^{\frac{5}{2}}} - \frac{a b c}{2 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a b \sqrt{d}}{4 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a b d^{\frac{3}{2}}}{4 c x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{4 c^{\frac{3}{2}}} - \frac{b^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} - \frac{b^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2 \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**7,x)

[Out]

-a**2*c/(6*sqrt(d)*x**7*sqrt(c/(d*x**2) + 1)) - 5*a**2*sqrt(d)/(24*x**5*sqrt(c/(
d*x**2) + 1)) + a**2*d**(3/2)/(48*c*x**3*sqrt(c/(d*x**2) + 1)) + a**2*d**(5/2)/(
16*c**2*x*sqrt(c/(d*x**2) + 1)) - a**2*d**3*asinh(sqrt(c)/(sqrt(d)*x))/(16*c**(5
/2)) - a*b*c/(2*sqrt(d)*x**5*sqrt(c/(d*x**2) + 1)) - 3*a*b*sqrt(d)/(4*x**3*sqrt(
c/(d*x**2) + 1)) - a*b*d**(3/2)/(4*c*x*sqrt(c/(d*x**2) + 1)) + a*b*d**2*asinh(sq
rt(c)/(sqrt(d)*x))/(4*c**(3/2)) - b**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(2*x) - b**2
*d*asinh(sqrt(c)/(sqrt(d)*x))/(2*sqrt(c))

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GIAC/XCAS [A]  time = 0.24728, size = 300, normalized size = 2.01 \[ \frac{\frac{3 \,{\left (8 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} + a^{2} d^{4}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2}} - \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c^{2} d^{2} - 48 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt{d x^{2} + c} b^{2} c^{4} d^{2} + 12 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b c d^{3} - 12 \, \sqrt{d x^{2} + c} a b c^{3} d^{3} - 3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{4} + 8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{4} + 3 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{4}}{c^{2} d^{3} x^{6}}}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^7,x, algorithm="giac")

[Out]

1/48*(3*(8*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*arctan(sqrt(d*x^2 + c)/sqrt(-c))
/(sqrt(-c)*c^2) - (24*(d*x^2 + c)^(5/2)*b^2*c^2*d^2 - 48*(d*x^2 + c)^(3/2)*b^2*c
^3*d^2 + 24*sqrt(d*x^2 + c)*b^2*c^4*d^2 + 12*(d*x^2 + c)^(5/2)*a*b*c*d^3 - 12*sq
rt(d*x^2 + c)*a*b*c^3*d^3 - 3*(d*x^2 + c)^(5/2)*a^2*d^4 + 8*(d*x^2 + c)^(3/2)*a^
2*c*d^4 + 3*sqrt(d*x^2 + c)*a^2*c^2*d^4)/(c^2*d^3*x^6))/d