3.634 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}}{x^6} \, dx\)

Optimal. Leaf size=228 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac{\left (c+d x^2\right )^{5/2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{15 c^2 x}+\frac{d x \left (c+d x^2\right )^{3/2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{12 c^2}+\frac{d x \sqrt{c+d x^2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{8 c}+\frac{1}{8} \sqrt{d} \left (8 a d (a d+5 b c)+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )-\frac{2 a \left (c+d x^2\right )^{7/2} (a d+5 b c)}{15 c^2 x^3} \]

[Out]

(d*(15*b^2*c^2 + 8*a*d*(5*b*c + a*d))*x*Sqrt[c + d*x^2])/(8*c) + (d*(15*b^2*c^2
+ 8*a*d*(5*b*c + a*d))*x*(c + d*x^2)^(3/2))/(12*c^2) - ((15*b^2*c^2 + 8*a*d*(5*b
*c + a*d))*(c + d*x^2)^(5/2))/(15*c^2*x) - (a^2*(c + d*x^2)^(7/2))/(5*c*x^5) - (
2*a*(5*b*c + a*d)*(c + d*x^2)^(7/2))/(15*c^2*x^3) + (Sqrt[d]*(15*b^2*c^2 + 8*a*d
*(5*b*c + a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/8

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Rubi [A]  time = 0.424608, antiderivative size = 225, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{a^2 \left (c+d x^2\right )^{7/2}}{5 c x^5}-\frac{\left (c+d x^2\right )^{5/2} \left (\frac{8 a d (a d+5 b c)}{c^2}+15 b^2\right )}{15 x}+\frac{d x \left (c+d x^2\right )^{3/2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{12 c^2}+\frac{d x \sqrt{c+d x^2} \left (8 a d (a d+5 b c)+15 b^2 c^2\right )}{8 c}+\frac{1}{8} \sqrt{d} \left (8 a d (a d+5 b c)+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )-\frac{2 a \left (c+d x^2\right )^{7/2} (a d+5 b c)}{15 c^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^6,x]

[Out]

(d*(15*b^2*c^2 + 8*a*d*(5*b*c + a*d))*x*Sqrt[c + d*x^2])/(8*c) + (d*(15*b^2*c^2
+ 8*a*d*(5*b*c + a*d))*x*(c + d*x^2)^(3/2))/(12*c^2) - ((15*b^2 + (8*a*d*(5*b*c
+ a*d))/c^2)*(c + d*x^2)^(5/2))/(15*x) - (a^2*(c + d*x^2)^(7/2))/(5*c*x^5) - (2*
a*(5*b*c + a*d)*(c + d*x^2)^(7/2))/(15*c^2*x^3) + (Sqrt[d]*(15*b^2*c^2 + 8*a*d*(
5*b*c + a*d))*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/8

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Rubi in Sympy [A]  time = 33.5802, size = 216, normalized size = 0.95 \[ - \frac{a^{2} \left (c + d x^{2}\right )^{\frac{7}{2}}}{5 c x^{5}} - \frac{2 a \left (c + d x^{2}\right )^{\frac{7}{2}} \left (a d + 5 b c\right )}{15 c^{2} x^{3}} + \frac{\sqrt{d} \left (8 a d \left (a d + 5 b c\right ) + 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} x}{\sqrt{c + d x^{2}}} \right )}}{8} + \frac{d x \sqrt{c + d x^{2}} \left (8 a d \left (a d + 5 b c\right ) + 15 b^{2} c^{2}\right )}{8 c} + \frac{d x \left (c + d x^{2}\right )^{\frac{3}{2}} \left (8 a d \left (a d + 5 b c\right ) + 15 b^{2} c^{2}\right )}{12 c^{2}} - \frac{\left (c + d x^{2}\right )^{\frac{5}{2}} \left (8 a d \left (a d + 5 b c\right ) + 15 b^{2} c^{2}\right )}{15 c^{2} x} \]

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**6,x)

[Out]

-a**2*(c + d*x**2)**(7/2)/(5*c*x**5) - 2*a*(c + d*x**2)**(7/2)*(a*d + 5*b*c)/(15
*c**2*x**3) + sqrt(d)*(8*a*d*(a*d + 5*b*c) + 15*b**2*c**2)*atanh(sqrt(d)*x/sqrt(
c + d*x**2))/8 + d*x*sqrt(c + d*x**2)*(8*a*d*(a*d + 5*b*c) + 15*b**2*c**2)/(8*c)
+ d*x*(c + d*x**2)**(3/2)*(8*a*d*(a*d + 5*b*c) + 15*b**2*c**2)/(12*c**2) - (c +
d*x**2)**(5/2)*(8*a*d*(a*d + 5*b*c) + 15*b**2*c**2)/(15*c**2*x)

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Mathematica [A]  time = 0.301412, size = 158, normalized size = 0.69 \[ \frac{1}{8} \sqrt{d} \left (8 a^2 d^2+40 a b c d+15 b^2 c^2\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )+\sqrt{c+d x^2} \left (\frac{-23 a^2 d^2-70 a b c d-15 b^2 c^2}{15 x}-\frac{a^2 c^2}{5 x^5}-\frac{a c (11 a d+10 b c)}{15 x^3}+\frac{1}{8} b d x (8 a d+9 b c)+\frac{1}{4} b^2 d^2 x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*(c + d*x^2)^(5/2))/x^6,x]

[Out]

Sqrt[c + d*x^2]*(-(a^2*c^2)/(5*x^5) - (a*c*(10*b*c + 11*a*d))/(15*x^3) + (-15*b^
2*c^2 - 70*a*b*c*d - 23*a^2*d^2)/(15*x) + (b*d*(9*b*c + 8*a*d)*x)/8 + (b^2*d^2*x
^3)/4) + (Sqrt[d]*(15*b^2*c^2 + 40*a*b*c*d + 8*a^2*d^2)*Log[d*x + Sqrt[d]*Sqrt[c
+ d*x^2]])/8

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Maple [A]  time = 0.024, size = 369, normalized size = 1.6 \[ -{\frac{{a}^{2}}{5\,c{x}^{5}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{2\,{a}^{2}d}{15\,{c}^{2}{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{8\,{a}^{2}{d}^{2}}{15\,{c}^{3}x} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{8\,{a}^{2}{d}^{3}x}{15\,{c}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{2\,{a}^{2}{d}^{3}x}{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{{a}^{2}{d}^{3}x}{c}\sqrt{d{x}^{2}+c}}+{a}^{2}{d}^{{\frac{5}{2}}}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) -{\frac{{b}^{2}}{cx} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{{b}^{2}dx}{c} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{5\,{b}^{2}dx}{4} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+{\frac{15\,{b}^{2}dcx}{8}\sqrt{d{x}^{2}+c}}+{\frac{15\,{b}^{2}{c}^{2}}{8}\sqrt{d}\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) }-{\frac{2\,ab}{3\,c{x}^{3}} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}-{\frac{8\,abd}{3\,{c}^{2}x} \left ( d{x}^{2}+c \right ) ^{{\frac{7}{2}}}}+{\frac{8\,ab{d}^{2}x}{3\,{c}^{2}} \left ( d{x}^{2}+c \right ) ^{{\frac{5}{2}}}}+{\frac{10\,ab{d}^{2}x}{3\,c} \left ( d{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+5\,ab{d}^{2}x\sqrt{d{x}^{2}+c}+5\,ab{d}^{3/2}c\ln \left ( x\sqrt{d}+\sqrt{d{x}^{2}+c} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(5/2)/x^6,x)

[Out]

-1/5*a^2*(d*x^2+c)^(7/2)/c/x^5-2/15*a^2*d/c^2/x^3*(d*x^2+c)^(7/2)-8/15*a^2*d^2/c
^3/x*(d*x^2+c)^(7/2)+8/15*a^2*d^3/c^3*x*(d*x^2+c)^(5/2)+2/3*a^2*d^3/c^2*x*(d*x^2
+c)^(3/2)+a^2*d^3/c*x*(d*x^2+c)^(1/2)+a^2*d^(5/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-
b^2/c/x*(d*x^2+c)^(7/2)+b^2*d/c*x*(d*x^2+c)^(5/2)+5/4*b^2*d*x*(d*x^2+c)^(3/2)+15
/8*b^2*d*c*x*(d*x^2+c)^(1/2)+15/8*b^2*d^(1/2)*c^2*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-
2/3*a*b/c/x^3*(d*x^2+c)^(7/2)-8/3*a*b*d/c^2/x*(d*x^2+c)^(7/2)+8/3*a*b*d^2/c^2*x*
(d*x^2+c)^(5/2)+10/3*a*b*d^2/c*x*(d*x^2+c)^(3/2)+5*a*b*d^2*x*(d*x^2+c)^(1/2)+5*a
*b*d^(3/2)*c*ln(x*d^(1/2)+(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*(d*x^2 + c)^(5/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.343575, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (15 \, b^{2} c^{2} + 40 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt{d} x^{5} \log \left (-2 \, d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{d} x - c\right ) + 2 \,{\left (30 \, b^{2} d^{2} x^{8} + 15 \,{\left (9 \, b^{2} c d + 8 \, a b d^{2}\right )} x^{6} - 8 \,{\left (15 \, b^{2} c^{2} + 70 \, a b c d + 23 \, a^{2} d^{2}\right )} x^{4} - 24 \, a^{2} c^{2} - 8 \,{\left (10 \, a b c^{2} + 11 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{240 \, x^{5}}, \frac{15 \,{\left (15 \, b^{2} c^{2} + 40 \, a b c d + 8 \, a^{2} d^{2}\right )} \sqrt{-d} x^{5} \arctan \left (\frac{d x}{\sqrt{d x^{2} + c} \sqrt{-d}}\right ) +{\left (30 \, b^{2} d^{2} x^{8} + 15 \,{\left (9 \, b^{2} c d + 8 \, a b d^{2}\right )} x^{6} - 8 \,{\left (15 \, b^{2} c^{2} + 70 \, a b c d + 23 \, a^{2} d^{2}\right )} x^{4} - 24 \, a^{2} c^{2} - 8 \,{\left (10 \, a b c^{2} + 11 \, a^{2} c d\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{120 \, x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x^6,x, algorithm="fricas")

[Out]

[1/240*(15*(15*b^2*c^2 + 40*a*b*c*d + 8*a^2*d^2)*sqrt(d)*x^5*log(-2*d*x^2 - 2*sq
rt(d*x^2 + c)*sqrt(d)*x - c) + 2*(30*b^2*d^2*x^8 + 15*(9*b^2*c*d + 8*a*b*d^2)*x^
6 - 8*(15*b^2*c^2 + 70*a*b*c*d + 23*a^2*d^2)*x^4 - 24*a^2*c^2 - 8*(10*a*b*c^2 +
11*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/x^5, 1/120*(15*(15*b^2*c^2 + 40*a*b*c*d + 8*a^
2*d^2)*sqrt(-d)*x^5*arctan(d*x/(sqrt(d*x^2 + c)*sqrt(-d))) + (30*b^2*d^2*x^8 + 1
5*(9*b^2*c*d + 8*a*b*d^2)*x^6 - 8*(15*b^2*c^2 + 70*a*b*c*d + 23*a^2*d^2)*x^4 - 2
4*a^2*c^2 - 8*(10*a*b*c^2 + 11*a^2*c*d)*x^2)*sqrt(d*x^2 + c))/x^5]

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Sympy [A]  time = 48.1252, size = 474, normalized size = 2.08 \[ - \frac{a^{2} \sqrt{c} d^{2}}{x \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{a^{2} c^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{5 x^{4}} - \frac{11 a^{2} c d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15 x^{2}} - \frac{8 a^{2} d^{\frac{5}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{15} + a^{2} d^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} - \frac{a^{2} d^{3} x}{\sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} - \frac{4 a b c^{\frac{3}{2}} d}{x \sqrt{1 + \frac{d x^{2}}{c}}} + a b \sqrt{c} d^{2} x \sqrt{1 + \frac{d x^{2}}{c}} - \frac{4 a b \sqrt{c} d^{2} x}{\sqrt{1 + \frac{d x^{2}}{c}}} - \frac{2 a b c^{2} \sqrt{d} \sqrt{\frac{c}{d x^{2}} + 1}}{3 x^{2}} - \frac{2 a b c d^{\frac{3}{2}} \sqrt{\frac{c}{d x^{2}} + 1}}{3} + 5 a b c d^{\frac{3}{2}} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )} - \frac{b^{2} c^{\frac{5}{2}}}{x \sqrt{1 + \frac{d x^{2}}{c}}} + b^{2} c^{\frac{3}{2}} d x \sqrt{1 + \frac{d x^{2}}{c}} - \frac{7 b^{2} c^{\frac{3}{2}} d x}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{3 b^{2} \sqrt{c} d^{2} x^{3}}{8 \sqrt{1 + \frac{d x^{2}}{c}}} + \frac{15 b^{2} c^{2} \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8} + \frac{b^{2} d^{3} x^{5}}{4 \sqrt{c} \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(5/2)/x**6,x)

[Out]

-a**2*sqrt(c)*d**2/(x*sqrt(1 + d*x**2/c)) - a**2*c**2*sqrt(d)*sqrt(c/(d*x**2) +
1)/(5*x**4) - 11*a**2*c*d**(3/2)*sqrt(c/(d*x**2) + 1)/(15*x**2) - 8*a**2*d**(5/2
)*sqrt(c/(d*x**2) + 1)/15 + a**2*d**(5/2)*asinh(sqrt(d)*x/sqrt(c)) - a**2*d**3*x
/(sqrt(c)*sqrt(1 + d*x**2/c)) - 4*a*b*c**(3/2)*d/(x*sqrt(1 + d*x**2/c)) + a*b*sq
rt(c)*d**2*x*sqrt(1 + d*x**2/c) - 4*a*b*sqrt(c)*d**2*x/sqrt(1 + d*x**2/c) - 2*a*
b*c**2*sqrt(d)*sqrt(c/(d*x**2) + 1)/(3*x**2) - 2*a*b*c*d**(3/2)*sqrt(c/(d*x**2)
+ 1)/3 + 5*a*b*c*d**(3/2)*asinh(sqrt(d)*x/sqrt(c)) - b**2*c**(5/2)/(x*sqrt(1 + d
*x**2/c)) + b**2*c**(3/2)*d*x*sqrt(1 + d*x**2/c) - 7*b**2*c**(3/2)*d*x/(8*sqrt(1
+ d*x**2/c)) + 3*b**2*sqrt(c)*d**2*x**3/(8*sqrt(1 + d*x**2/c)) + 15*b**2*c**2*s
qrt(d)*asinh(sqrt(d)*x/sqrt(c))/8 + b**2*d**3*x**5/(4*sqrt(c)*sqrt(1 + d*x**2/c)
)

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GIAC/XCAS [A]  time = 0.25106, size = 689, normalized size = 3.02 \[ \frac{1}{8} \,{\left (2 \, b^{2} d^{2} x^{2} + \frac{9 \, b^{2} c d^{3} + 8 \, a b d^{4}}{d^{2}}\right )} \sqrt{d x^{2} + c} x - \frac{1}{16} \,{\left (15 \, b^{2} c^{2} \sqrt{d} + 40 \, a b c d^{\frac{3}{2}} + 8 \, a^{2} d^{\frac{5}{2}}\right )}{\rm ln}\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right ) + \frac{2 \,{\left (15 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} b^{2} c^{3} \sqrt{d} + 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a b c^{2} d^{\frac{3}{2}} + 45 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{8} a^{2} c d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} b^{2} c^{4} \sqrt{d} - 300 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a b c^{3} d^{\frac{3}{2}} - 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{6} a^{2} c^{2} d^{\frac{5}{2}} + 90 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b^{2} c^{5} \sqrt{d} + 400 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a b c^{4} d^{\frac{3}{2}} + 140 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} a^{2} c^{3} d^{\frac{5}{2}} - 60 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b^{2} c^{6} \sqrt{d} - 260 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c^{5} d^{\frac{3}{2}} - 70 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} c^{4} d^{\frac{5}{2}} + 15 \, b^{2} c^{7} \sqrt{d} + 70 \, a b c^{6} d^{\frac{3}{2}} + 23 \, a^{2} c^{5} d^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} - c\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^(5/2)/x^6,x, algorithm="giac")

[Out]

1/8*(2*b^2*d^2*x^2 + (9*b^2*c*d^3 + 8*a*b*d^4)/d^2)*sqrt(d*x^2 + c)*x - 1/16*(15
*b^2*c^2*sqrt(d) + 40*a*b*c*d^(3/2) + 8*a^2*d^(5/2))*ln((sqrt(d)*x - sqrt(d*x^2
+ c))^2) + 2/15*(15*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c^3*sqrt(d) + 90*(sqrt(d
)*x - sqrt(d*x^2 + c))^8*a*b*c^2*d^(3/2) + 45*(sqrt(d)*x - sqrt(d*x^2 + c))^8*a^
2*c*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c^4*sqrt(d) - 300*(sqrt(d)*
x - sqrt(d*x^2 + c))^6*a*b*c^3*d^(3/2) - 90*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a^2*
c^2*d^(5/2) + 90*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b^2*c^5*sqrt(d) + 400*(sqrt(d)*
x - sqrt(d*x^2 + c))^4*a*b*c^4*d^(3/2) + 140*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2
*c^3*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b^2*c^6*sqrt(d) - 260*(sqrt(d)
*x - sqrt(d*x^2 + c))^2*a*b*c^5*d^(3/2) - 70*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2
*c^4*d^(5/2) + 15*b^2*c^7*sqrt(d) + 70*a*b*c^6*d^(3/2) + 23*a^2*c^5*d^(5/2))/((s
qrt(d)*x - sqrt(d*x^2 + c))^2 - c)^5