3.659 \(\int \frac{\left (a+b x^2\right )^2}{x^7 \left (c+d x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=192 \[ -\frac{a^2}{6 c x^6 \sqrt{c+d x^2}}+\frac{d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{9/2}}-\frac{\sqrt{c+d x^2} \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 x^2}+\frac{24 b^2 c^2-5 a d (12 b c-7 a d)}{24 c^3 x^2 \sqrt{c+d x^2}}-\frac{a (12 b c-7 a d)}{24 c^2 x^4 \sqrt{c+d x^2}} \]

[Out]

-a^2/(6*c*x^6*Sqrt[c + d*x^2]) - (a*(12*b*c - 7*a*d))/(24*c^2*x^4*Sqrt[c + d*x^2
]) + (24*b^2*c^2 - 5*a*d*(12*b*c - 7*a*d))/(24*c^3*x^2*Sqrt[c + d*x^2]) - ((24*b
^2*c^2 - 5*a*d*(12*b*c - 7*a*d))*Sqrt[c + d*x^2])/(16*c^4*x^2) + (d*(24*b^2*c^2
- 5*a*d*(12*b*c - 7*a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(9/2))

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Rubi [A]  time = 0.526352, antiderivative size = 193, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{35 a^2 d^2-60 a b c d+24 b^2 c^2}{24 c^3 x^2 \sqrt{c+d x^2}}-\frac{a^2}{6 c x^6 \sqrt{c+d x^2}}+\frac{d \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{9/2}}-\frac{\sqrt{c+d x^2} \left (24 b^2 c^2-5 a d (12 b c-7 a d)\right )}{16 c^4 x^2}-\frac{a (12 b c-7 a d)}{24 c^2 x^4 \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x]

[Out]

-a^2/(6*c*x^6*Sqrt[c + d*x^2]) - (a*(12*b*c - 7*a*d))/(24*c^2*x^4*Sqrt[c + d*x^2
]) + (24*b^2*c^2 - 60*a*b*c*d + 35*a^2*d^2)/(24*c^3*x^2*Sqrt[c + d*x^2]) - ((24*
b^2*c^2 - 5*a*d*(12*b*c - 7*a*d))*Sqrt[c + d*x^2])/(16*c^4*x^2) + (d*(24*b^2*c^2
- 5*a*d*(12*b*c - 7*a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(9/2))

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Rubi in Sympy [A]  time = 35.4833, size = 184, normalized size = 0.96 \[ - \frac{a^{2}}{6 c x^{6} \sqrt{c + d x^{2}}} + \frac{a \left (7 a d - 12 b c\right )}{24 c^{2} x^{4} \sqrt{c + d x^{2}}} + \frac{5 a d \left (7 a d - 12 b c\right ) + 24 b^{2} c^{2}}{24 c^{3} x^{2} \sqrt{c + d x^{2}}} - \frac{\sqrt{c + d x^{2}} \left (5 a d \left (7 a d - 12 b c\right ) + 24 b^{2} c^{2}\right )}{16 c^{4} x^{2}} + \frac{d \left (5 a d \left (7 a d - 12 b c\right ) + 24 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{16 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x**7/(d*x**2+c)**(3/2),x)

[Out]

-a**2/(6*c*x**6*sqrt(c + d*x**2)) + a*(7*a*d - 12*b*c)/(24*c**2*x**4*sqrt(c + d*
x**2)) + (5*a*d*(7*a*d - 12*b*c) + 24*b**2*c**2)/(24*c**3*x**2*sqrt(c + d*x**2))
- sqrt(c + d*x**2)*(5*a*d*(7*a*d - 12*b*c) + 24*b**2*c**2)/(16*c**4*x**2) + d*(
5*a*d*(7*a*d - 12*b*c) + 24*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(16*c**(9
/2))

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Mathematica [A]  time = 0.558826, size = 190, normalized size = 0.99 \[ \frac{3 d \left (35 a^2 d^2-60 a b c d+24 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )-3 d \log (x) \left (35 a^2 d^2-60 a b c d+24 b^2 c^2\right )-\frac{\sqrt{c} \left (a^2 \left (8 c^3-14 c^2 d x^2+35 c d^2 x^4+105 d^3 x^6\right )+12 a b c x^2 \left (2 c^2-5 c d x^2-15 d^2 x^4\right )+24 b^2 c^2 x^4 \left (c+3 d x^2\right )\right )}{x^6 \sqrt{c+d x^2}}}{48 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/(x^7*(c + d*x^2)^(3/2)),x]

[Out]

(-((Sqrt[c]*(24*b^2*c^2*x^4*(c + 3*d*x^2) + 12*a*b*c*x^2*(2*c^2 - 5*c*d*x^2 - 15
*d^2*x^4) + a^2*(8*c^3 - 14*c^2*d*x^2 + 35*c*d^2*x^4 + 105*d^3*x^6)))/(x^6*Sqrt[
c + d*x^2])) - 3*d*(24*b^2*c^2 - 60*a*b*c*d + 35*a^2*d^2)*Log[x] + 3*d*(24*b^2*c
^2 - 60*a*b*c*d + 35*a^2*d^2)*Log[c + Sqrt[c]*Sqrt[c + d*x^2]])/(48*c^(9/2))

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Maple [A]  time = 0.02, size = 281, normalized size = 1.5 \[ -{\frac{{a}^{2}}{6\,c{x}^{6}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{7\,{a}^{2}d}{24\,{c}^{2}{x}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{35\,{a}^{2}{d}^{2}}{48\,{c}^{3}{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{35\,{a}^{2}{d}^{3}}{16\,{c}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{35\,{a}^{2}{d}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{9}{2}}}}-{\frac{{b}^{2}}{2\,c{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{3\,{b}^{2}d}{2\,{c}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{3\,{b}^{2}d}{2}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{5}{2}}}}-{\frac{ab}{2\,c{x}^{4}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{5\,abd}{4\,{c}^{2}{x}^{2}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}+{\frac{15\,ab{d}^{2}}{4\,{c}^{3}}{\frac{1}{\sqrt{d{x}^{2}+c}}}}-{\frac{15\,ab{d}^{2}}{4}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^7/(d*x^2+c)^(3/2),x)

[Out]

-1/6*a^2/c/x^6/(d*x^2+c)^(1/2)+7/24*a^2*d/c^2/x^4/(d*x^2+c)^(1/2)-35/48*a^2*d^2/
c^3/x^2/(d*x^2+c)^(1/2)-35/16*a^2*d^3/c^4/(d*x^2+c)^(1/2)+35/16*a^2*d^3/c^(9/2)*
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)-3/2*b^2*d/c^
2/(d*x^2+c)^(1/2)+3/2*b^2*d/c^(5/2)*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)+5/4*a*b*d/c^2/x^2/(d*x^2+c)^(1/2)+15/4*a*b*d^2/c^3/(d*x^
2+c)^(1/2)-15/4*a*b*d^2/c^(7/2)*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/((d*x^2 + c)^(3/2)*x^7),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247037, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (3 \,{\left (24 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} + 8 \, a^{2} c^{3} +{\left (24 \, b^{2} c^{3} - 60 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} + 2 \,{\left (12 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c} - 3 \,{\left ({\left (24 \, b^{2} c^{2} d^{2} - 60 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} +{\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6}\right )} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right )}{96 \,{\left (c^{4} d x^{8} + c^{5} x^{6}\right )} \sqrt{c}}, -\frac{{\left (3 \,{\left (24 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} x^{6} + 8 \, a^{2} c^{3} +{\left (24 \, b^{2} c^{3} - 60 \, a b c^{2} d + 35 \, a^{2} c d^{2}\right )} x^{4} + 2 \,{\left (12 \, a b c^{3} - 7 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c} - 3 \,{\left ({\left (24 \, b^{2} c^{2} d^{2} - 60 \, a b c d^{3} + 35 \, a^{2} d^{4}\right )} x^{8} +{\left (24 \, b^{2} c^{3} d - 60 \, a b c^{2} d^{2} + 35 \, a^{2} c d^{3}\right )} x^{6}\right )} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right )}{48 \,{\left (c^{4} d x^{8} + c^{5} x^{6}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^7),x, algorithm="fricas")

[Out]

[-1/96*(2*(3*(24*b^2*c^2*d - 60*a*b*c*d^2 + 35*a^2*d^3)*x^6 + 8*a^2*c^3 + (24*b^
2*c^3 - 60*a*b*c^2*d + 35*a^2*c*d^2)*x^4 + 2*(12*a*b*c^3 - 7*a^2*c^2*d)*x^2)*sqr
t(d*x^2 + c)*sqrt(c) - 3*((24*b^2*c^2*d^2 - 60*a*b*c*d^3 + 35*a^2*d^4)*x^8 + (24
*b^2*c^3*d - 60*a*b*c^2*d^2 + 35*a^2*c*d^3)*x^6)*log(-((d*x^2 + 2*c)*sqrt(c) + 2
*sqrt(d*x^2 + c)*c)/x^2))/((c^4*d*x^8 + c^5*x^6)*sqrt(c)), -1/48*((3*(24*b^2*c^2
*d - 60*a*b*c*d^2 + 35*a^2*d^3)*x^6 + 8*a^2*c^3 + (24*b^2*c^3 - 60*a*b*c^2*d + 3
5*a^2*c*d^2)*x^4 + 2*(12*a*b*c^3 - 7*a^2*c^2*d)*x^2)*sqrt(d*x^2 + c)*sqrt(-c) -
3*((24*b^2*c^2*d^2 - 60*a*b*c*d^3 + 35*a^2*d^4)*x^8 + (24*b^2*c^3*d - 60*a*b*c^2
*d^2 + 35*a^2*c*d^3)*x^6)*arctan(sqrt(-c)/sqrt(d*x^2 + c)))/((c^4*d*x^8 + c^5*x^
6)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2}\right )^{2}}{x^{7} \left (c + d x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**7/(d*x**2+c)**(3/2),x)

[Out]

Integral((a + b*x**2)**2/(x**7*(c + d*x**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.240611, size = 360, normalized size = 1.88 \[ -\frac{{\left (24 \, b^{2} c^{2} d - 60 \, a b c d^{2} + 35 \, a^{2} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{16 \, \sqrt{-c} c^{4}} - \frac{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}}{\sqrt{d x^{2} + c} c^{4}} - \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c^{2} d - 48 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{3} d + 24 \, \sqrt{d x^{2} + c} b^{2} c^{4} d - 84 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b c d^{2} + 192 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c^{2} d^{2} - 108 \, \sqrt{d x^{2} + c} a b c^{3} d^{2} + 57 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{3} - 136 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{3} + 87 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{3}}{48 \, c^{4} d^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2/((d*x^2 + c)^(3/2)*x^7),x, algorithm="giac")

[Out]

-1/16*(24*b^2*c^2*d - 60*a*b*c*d^2 + 35*a^2*d^3)*arctan(sqrt(d*x^2 + c)/sqrt(-c)
)/(sqrt(-c)*c^4) - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)/(sqrt(d*x^2 + c)*c^4) - 1
/48*(24*(d*x^2 + c)^(5/2)*b^2*c^2*d - 48*(d*x^2 + c)^(3/2)*b^2*c^3*d + 24*sqrt(d
*x^2 + c)*b^2*c^4*d - 84*(d*x^2 + c)^(5/2)*a*b*c*d^2 + 192*(d*x^2 + c)^(3/2)*a*b
*c^2*d^2 - 108*sqrt(d*x^2 + c)*a*b*c^3*d^2 + 57*(d*x^2 + c)^(5/2)*a^2*d^3 - 136*
(d*x^2 + c)^(3/2)*a^2*c*d^3 + 87*sqrt(d*x^2 + c)*a^2*c^2*d^3)/(c^4*d^3*x^6)