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

Optimal. Leaf size=151 \[ \frac{d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{7/2}}-\frac{\sqrt{c+d x^2} \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right )}{16 c^3 x^2}-\frac{a^2 \sqrt{c+d x^2}}{6 c x^6}-\frac{a \sqrt{c+d x^2} (12 b c-5 a d)}{24 c^2 x^4} \]

[Out]

-(a^2*Sqrt[c + d*x^2])/(6*c*x^6) - (a*(12*b*c - 5*a*d)*Sqrt[c + d*x^2])/(24*c^2*
x^4) - ((8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x^2])/(16*c^3*x^2) + (d*
(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(7/
2))

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Rubi [A]  time = 0.399052, antiderivative size = 151, 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 \[ \frac{d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{7/2}}-\frac{\sqrt{c+d x^2} \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right )}{16 c^3 x^2}-\frac{a^2 \sqrt{c+d x^2}}{6 c x^6}-\frac{a \sqrt{c+d x^2} (12 b c-5 a d)}{24 c^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*Sqrt[c + d*x^2])/(6*c*x^6) - (a*(12*b*c - 5*a*d)*Sqrt[c + d*x^2])/(24*c^2*
x^4) - ((8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x^2])/(16*c^3*x^2) + (d*
(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(7/
2))

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Rubi in Sympy [A]  time = 29.7328, size = 138, normalized size = 0.91 \[ - \frac{a^{2} \sqrt{c + d x^{2}}}{6 c x^{6}} + \frac{a \sqrt{c + d x^{2}} \left (5 a d - 12 b c\right )}{24 c^{2} x^{4}} - \frac{\sqrt{c + d x^{2}} \left (a d \left (5 a d - 12 b c\right ) + 8 b^{2} c^{2}\right )}{16 c^{3} x^{2}} + \frac{d \left (a d \left (5 a d - 12 b c\right ) + 8 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{16 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(c + d*x**2)/(6*c*x**6) + a*sqrt(c + d*x**2)*(5*a*d - 12*b*c)/(24*c**2
*x**4) - sqrt(c + d*x**2)*(a*d*(5*a*d - 12*b*c) + 8*b**2*c**2)/(16*c**3*x**2) +
d*(a*d*(5*a*d - 12*b*c) + 8*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(16*c**(7
/2))

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Mathematica [A]  time = 0.224995, size = 168, normalized size = 1.11 \[ \frac{d \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{16 c^{7/2}}-\frac{d \log (x) \left (5 a^2 d^2-12 a b c d+8 b^2 c^2\right )}{16 c^{7/2}}+\sqrt{c+d x^2} \left (\frac{-5 a^2 d^2+12 a b c d-8 b^2 c^2}{16 c^3 x^2}-\frac{a^2}{6 c x^6}+\frac{a (5 a d-12 b c)}{24 c^2 x^4}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-a^2/(6*c*x^6) + (a*(-12*b*c + 5*a*d))/(24*c^2*x^4) + (-8*b^2*c^2 + 12*a*b*c*d
- 5*a^2*d^2)/(16*c^3*x^2))*Sqrt[c + d*x^2] - (d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*
d^2)*Log[x])/(16*c^(7/2)) + (d*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*Log[c + Sqrt
[c]*Sqrt[c + d*x^2]])/(16*c^(7/2))

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Maple [A]  time = 0.019, size = 224, normalized size = 1.5 \[ -{\frac{{a}^{2}}{6\,c{x}^{6}}\sqrt{d{x}^{2}+c}}+{\frac{5\,{a}^{2}d}{24\,{c}^{2}{x}^{4}}\sqrt{d{x}^{2}+c}}-{\frac{5\,{a}^{2}{d}^{2}}{16\,{c}^{3}{x}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{5\,{a}^{2}{d}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{7}{2}}}}-{\frac{{b}^{2}}{2\,c{x}^{2}}\sqrt{d{x}^{2}+c}}+{\frac{{b}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{ab}{2\,c{x}^{4}}\sqrt{d{x}^{2}+c}}+{\frac{3\,abd}{4\,{c}^{2}{x}^{2}}\sqrt{d{x}^{2}+c}}-{\frac{3\,ab{d}^{2}}{4}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*a^2*(d*x^2+c)^(1/2)/c/x^6+5/24*a^2*d/c^2/x^4*(d*x^2+c)^(1/2)-5/16*a^2*d^2/c
^3/x^2*(d*x^2+c)^(1/2)+5/16*a^2*d^3/c^(7/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x
)-1/2*b^2/c/x^2*(d*x^2+c)^(1/2)+1/2*b^2*d/c^(3/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1
/2))/x)-1/2*a*b/c/x^4*(d*x^2+c)^(1/2)+3/4*a*b*d/c^2/x^2*(d*x^2+c)^(1/2)-3/4*a*b*
d^2/c^(5/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)

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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 = 0.294137, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (8 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{6} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (8 \, b^{2} c^{2} - 12 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 2 \,{\left (12 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c}}{96 \, c^{\frac{7}{2}} x^{6}}, \frac{3 \,{\left (8 \, b^{2} c^{2} d - 12 \, a b c d^{2} + 5 \, a^{2} d^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (3 \,{\left (8 \, b^{2} c^{2} - 12 \, a b c d + 5 \, a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 2 \,{\left (12 \, a b c^{2} - 5 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{48 \, \sqrt{-c} c^{3} x^{6}}\right ] \]

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 - 12*a*b*c*d^2 + 5*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 - 12*a*b*c*d + 5*a^2*d^2)*x^4 +
8*a^2*c^2 + 2*(12*a*b*c^2 - 5*a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(c))/(c^(7/2)*x^
6), 1/48*(3*(8*b^2*c^2*d - 12*a*b*c*d^2 + 5*a^2*d^3)*x^6*arctan(sqrt(-c)/sqrt(d*
x^2 + c)) - (3*(8*b^2*c^2 - 12*a*b*c*d + 5*a^2*d^2)*x^4 + 8*a^2*c^2 + 2*(12*a*b*
c^2 - 5*a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(-c))/(sqrt(-c)*c^3*x^6)]

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Sympy [A]  time = 130.835, size = 301, normalized size = 1.99 \[ - \frac{a^{2}}{6 \sqrt{d} x^{7} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} \sqrt{d}}{24 c x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{5 a^{2} d^{\frac{3}{2}}}{48 c^{2} x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{5 a^{2} d^{\frac{5}{2}}}{16 c^{3} x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{5 a^{2} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{16 c^{\frac{7}{2}}} - \frac{a b}{2 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a b \sqrt{d}}{4 c x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{3 a b d^{\frac{3}{2}}}{4 c^{2} x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{4 c^{\frac{5}{2}}} - \frac{b^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 c x} + \frac{b^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.234078, size = 325, normalized size = 2.15 \[ -\frac{\frac{3 \,{\left (8 \, b^{2} c^{2} d^{2} - 12 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}} + \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c^{2} d^{2} - 48 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt{d x^{2} + c} b^{2} c^{4} d^{2} - 36 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b c d^{3} + 96 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c^{2} d^{3} - 60 \, \sqrt{d x^{2} + c} a b c^{3} d^{3} + 15 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{4} - 40 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{4} + 33 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{4}}{c^{3} d^{3} x^{6}}}{48 \, d} \]

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 - 12*a*b*c*d^3 + 5*a^2*d^4)*arctan(sqrt(d*x^2 + c)/sqrt(
-c))/(sqrt(-c)*c^3) + (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 - 36*(d*x^2 + c)^(5/2)*a*b*c*d^3 + 9
6*(d*x^2 + c)^(3/2)*a*b*c^2*d^3 - 60*sqrt(d*x^2 + c)*a*b*c^3*d^3 + 15*(d*x^2 + c
)^(5/2)*a^2*d^4 - 40*(d*x^2 + c)^(3/2)*a^2*c*d^4 + 33*sqrt(d*x^2 + c)*a^2*c^2*d^
4)/(c^3*d^3*x^6))/d