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

Optimal. Leaf size=187 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{\left (c+d x^2\right )^{3/2} \left (a d (12 b c-a d)+24 b^2 c^2\right )}{48 c^2 x^2}+\frac{d \sqrt{c+d x^2} \left (a d (12 b c-a d)+24 b^2 c^2\right )}{16 c^2}-\frac{d \left (a d (12 b c-a d)+24 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{3/2}}-\frac{a \left (c+d x^2\right )^{5/2} (12 b c-a d)}{24 c^2 x^4} \]

[Out]

(d*(24*b^2*c^2 + a*d*(12*b*c - a*d))*Sqrt[c + d*x^2])/(16*c^2) - ((24*b^2*c^2 +
a*d*(12*b*c - a*d))*(c + d*x^2)^(3/2))/(48*c^2*x^2) - (a^2*(c + d*x^2)^(5/2))/(6
*c*x^6) - (a*(12*b*c - a*d)*(c + d*x^2)^(5/2))/(24*c^2*x^4) - (d*(24*b^2*c^2 + a
*d*(12*b*c - a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(3/2))

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Rubi [A]  time = 0.527798, antiderivative size = 184, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{a^2 \left (c+d x^2\right )^{5/2}}{6 c x^6}-\frac{\left (c+d x^2\right )^{3/2} \left (\frac{a d (12 b c-a d)}{c^2}+24 b^2\right )}{48 x^2}+\frac{d \sqrt{c+d x^2} \left (a d (12 b c-a d)+24 b^2 c^2\right )}{16 c^2}-\frac{d \left (a d (12 b c-a d)+24 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{16 c^{3/2}}-\frac{a \left (c+d x^2\right )^{5/2} (12 b c-a d)}{24 c^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(d*(24*b^2*c^2 + a*d*(12*b*c - a*d))*Sqrt[c + d*x^2])/(16*c^2) - ((24*b^2 + (a*d
*(12*b*c - a*d))/c^2)*(c + d*x^2)^(3/2))/(48*x^2) - (a^2*(c + d*x^2)^(5/2))/(6*c
*x^6) - (a*(12*b*c - a*d)*(c + d*x^2)^(5/2))/(24*c^2*x^4) - (d*(24*b^2*c^2 + a*d
*(12*b*c - a*d))*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/(16*c^(3/2))

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Rubi in Sympy [A]  time = 34.1998, size = 170, normalized size = 0.91 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{5}{2}}}{6 c x^{6}} + \frac{a \left (c + d x^{2}\right )^{\frac{5}{2}} \left (a d - 12 b c\right )}{24 c^{2} x^{4}} + \frac{d \sqrt{c + d x^{2}} \left (- a d \left (a d - 12 b c\right ) + 24 b^{2} c^{2}\right )}{16 c^{2}} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (- a d \left (a d - 12 b c\right ) + 24 b^{2} c^{2}\right )}{48 c^{2} x^{2}} - \frac{d \left (- a d \left (a d - 12 b c\right ) + 24 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{16 c^{\frac{3}{2}}} \]

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**7,x)

[Out]

-a**2*(c + d*x**2)**(5/2)/(6*c*x**6) + a*(c + d*x**2)**(5/2)*(a*d - 12*b*c)/(24*
c**2*x**4) + d*sqrt(c + d*x**2)*(-a*d*(a*d - 12*b*c) + 24*b**2*c**2)/(16*c**2) -
(c + d*x**2)**(3/2)*(-a*d*(a*d - 12*b*c) + 24*b**2*c**2)/(48*c**2*x**2) - d*(-a
*d*(a*d - 12*b*c) + 24*b**2*c**2)*atanh(sqrt(c + d*x**2)/sqrt(c))/(16*c**(3/2))

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Mathematica [A]  time = 0.329896, size = 166, normalized size = 0.89 \[ \sqrt{c+d x^2} \left (\frac{-a^2 d^2-20 a b c d-8 b^2 c^2}{16 c x^2}-\frac{a^2 c}{6 x^6}-\frac{a (7 a d+12 b c)}{24 x^4}+b^2 d\right )+\frac{d \left (a^2 d^2-12 a b c d-24 b^2 c^2\right ) \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )}{16 c^{3/2}}-\frac{d \log (x) \left (a^2 d^2-12 a b c d-24 b^2 c^2\right )}{16 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(b^2*d - (a^2*c)/(6*x^6) - (a*(12*b*c + 7*a*d))/(24*x^4) + (-8*b^2*c^2 - 20*a*b*
c*d - a^2*d^2)/(16*c*x^2))*Sqrt[c + d*x^2] - (d*(-24*b^2*c^2 - 12*a*b*c*d + a^2*
d^2)*Log[x])/(16*c^(3/2)) + (d*(-24*b^2*c^2 - 12*a*b*c*d + a^2*d^2)*Log[c + Sqrt
[c]*Sqrt[c + d*x^2]])/(16*c^(3/2))

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Maple [B]  time = 0.02, size = 335, normalized size = 1.8 \[ -{\frac{{a}^{2}}{6\,c{x}^{6}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}d}{24\,{c}^{2}{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{a}^{2}{d}^{2}}{48\,{c}^{3}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{{a}^{2}{d}^{3}}{48\,{c}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{d}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){c}^{-{\frac{3}{2}}}}-{\frac{{a}^{2}{d}^{3}}{16\,{c}^{2}}\sqrt{d{x}^{2}+c}}-{\frac{{b}^{2}}{2\,c{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{{b}^{2}d}{2\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}d}{2}\sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ) }+{\frac{3\,{b}^{2}d}{2}\sqrt{d{x}^{2}+c}}-{\frac{ab}{2\,c{x}^{4}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}-{\frac{abd}{4\,{c}^{2}{x}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{ab{d}^{2}}{4\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ab{d}^{2}}{4}\ln \left ({\frac{1}{x} \left ( 2\,c+2\,\sqrt{c}\sqrt{d{x}^{2}+c} \right ) } \right ){\frac{1}{\sqrt{c}}}}+{\frac{3\,ab{d}^{2}}{4\,c}\sqrt{d{x}^{2}+c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239103, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (24 \, b^{2} c^{2} d + 12 \, a b c d^{2} - a^{2} d^{3}\right )} x^{6} \log \left (-\frac{{\left (d x^{2} + 2 \, c\right )} \sqrt{c} + 2 \, \sqrt{d x^{2} + c} c}{x^{2}}\right ) - 2 \,{\left (48 \, b^{2} c d x^{6} - 3 \,{\left (8 \, b^{2} c^{2} + 20 \, a b c d + a^{2} d^{2}\right )} x^{4} - 8 \, a^{2} c^{2} - 2 \,{\left (12 \, a b c^{2} + 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{c}}{96 \, c^{\frac{3}{2}} x^{6}}, -\frac{3 \,{\left (24 \, b^{2} c^{2} d + 12 \, a b c d^{2} - a^{2} d^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-c}}{\sqrt{d x^{2} + c}}\right ) -{\left (48 \, b^{2} c d x^{6} - 3 \,{\left (8 \, b^{2} c^{2} + 20 \, a b c d + a^{2} d^{2}\right )} x^{4} - 8 \, a^{2} c^{2} - 2 \,{\left (12 \, a b c^{2} + 7 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c} \sqrt{-c}}{48 \, \sqrt{-c} c x^{6}}\right ] \]

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*(3*(24*b^2*c^2*d + 12*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*(48*b^2*c*d*x^6 - 3*(8*b^2*c^2 + 20*a*b*c*d +
a^2*d^2)*x^4 - 8*a^2*c^2 - 2*(12*a*b*c^2 + 7*a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(
c))/(c^(3/2)*x^6), -1/48*(3*(24*b^2*c^2*d + 12*a*b*c*d^2 - a^2*d^3)*x^6*arctan(s
qrt(-c)/sqrt(d*x^2 + c)) - (48*b^2*c*d*x^6 - 3*(8*b^2*c^2 + 20*a*b*c*d + a^2*d^2
)*x^4 - 8*a^2*c^2 - 2*(12*a*b*c^2 + 7*a^2*c*d)*x^2)*sqrt(d*x^2 + c)*sqrt(-c))/(s
qrt(-c)*c*x^6)]

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Sympy [A]  time = 134.915, size = 367, normalized size = 1.96 \[ - \frac{a^{2} c^{2}}{6 \sqrt{d} x^{7} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{11 a^{2} c \sqrt{d}}{24 x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{17 a^{2} d^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a^{2} d^{\frac{5}{2}}}{16 c x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{a^{2} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{16 c^{\frac{3}{2}}} - \frac{a b c^{2}}{2 \sqrt{d} x^{5} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a b c \sqrt{d}}{4 x^{3} \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{a b d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{x} - \frac{a b d^{\frac{3}{2}}}{4 x \sqrt{\frac{c}{d x^{2}} + 1}} - \frac{3 a b d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{4 \sqrt{c}} - \frac{3 b^{2} \sqrt{c} d \operatorname{asinh}{\left (\frac{\sqrt{c}}{\sqrt{d} x} \right )}}{2} - \frac{b^{2} c \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{2 x} + \frac{b^{2} c \sqrt{d}}{x \sqrt{\frac{c}{d x^{2}} + 1}} + \frac{b^{2} d^{\frac{3}{2}} x}{\sqrt{\frac{c}{d x^{2}} + 1}} \]

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)

[Out]

-a**2*c**2/(6*sqrt(d)*x**7*sqrt(c/(d*x**2) + 1)) - 11*a**2*c*sqrt(d)/(24*x**5*sq
rt(c/(d*x**2) + 1)) - 17*a**2*d**(3/2)/(48*x**3*sqrt(c/(d*x**2) + 1)) - a**2*d**
(5/2)/(16*c*x*sqrt(c/(d*x**2) + 1)) + a**2*d**3*asinh(sqrt(c)/(sqrt(d)*x))/(16*c
**(3/2)) - a*b*c**2/(2*sqrt(d)*x**5*sqrt(c/(d*x**2) + 1)) - 3*a*b*c*sqrt(d)/(4*x
**3*sqrt(c/(d*x**2) + 1)) - a*b*d**(3/2)*sqrt(c/(d*x**2) + 1)/x - a*b*d**(3/2)/(
4*x*sqrt(c/(d*x**2) + 1)) - 3*a*b*d**2*asinh(sqrt(c)/(sqrt(d)*x))/(4*sqrt(c)) -
3*b**2*sqrt(c)*d*asinh(sqrt(c)/(sqrt(d)*x))/2 - b**2*c*sqrt(d)*sqrt(c/(d*x**2) +
1)/(2*x) + b**2*c*sqrt(d)/(x*sqrt(c/(d*x**2) + 1)) + b**2*d**(3/2)*x/sqrt(c/(d*
x**2) + 1)

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GIAC/XCAS [A]  time = 0.241855, size = 350, normalized size = 1.87 \[ \frac{48 \, \sqrt{d x^{2} + c} b^{2} d^{2} + \frac{3 \,{\left (24 \, b^{2} c^{2} d^{2} + 12 \, a b c d^{3} - a^{2} d^{4}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c} - \frac{24 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{2} c^{2} d^{2} - 48 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{2} c^{3} d^{2} + 24 \, \sqrt{d x^{2} + c} b^{2} c^{4} d^{2} + 60 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b c d^{3} - 96 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b c^{2} d^{3} + 36 \, \sqrt{d x^{2} + c} a b c^{3} d^{3} + 3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a^{2} d^{4} + 8 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} c d^{4} - 3 \, \sqrt{d x^{2} + c} a^{2} c^{2} d^{4}}{c d^{3} x^{6}}}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(48*sqrt(d*x^2 + c)*b^2*d^2 + 3*(24*b^2*c^2*d^2 + 12*a*b*c*d^3 - a^2*d^4)*a
rctan(sqrt(d*x^2 + c)/sqrt(-c))/(sqrt(-c)*c) - (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 + 60*(d*x^2
+ c)^(5/2)*a*b*c*d^3 - 96*(d*x^2 + c)^(3/2)*a*b*c^2*d^3 + 36*sqrt(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*d^3*x^6))/d