3.551 \(\int \frac{\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^8} \, dx\)

Optimal. Leaf size=108 \[ -\frac{A \left (a+b x^2\right )^{7/2}}{7 a x^7}+b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b^2 B \sqrt{a+b x^2}}{x}-\frac{B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b B \left (a+b x^2\right )^{3/2}}{3 x^3} \]

[Out]

-((b^2*B*Sqrt[a + b*x^2])/x) - (b*B*(a + b*x^2)^(3/2))/(3*x^3) - (B*(a + b*x^2)^
(5/2))/(5*x^5) - (A*(a + b*x^2)^(7/2))/(7*a*x^7) + b^(5/2)*B*ArcTanh[(Sqrt[b]*x)
/Sqrt[a + b*x^2]]

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Rubi [A]  time = 0.138362, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{A \left (a+b x^2\right )^{7/2}}{7 a x^7}+b^{5/2} B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )-\frac{b^2 B \sqrt{a+b x^2}}{x}-\frac{B \left (a+b x^2\right )^{5/2}}{5 x^5}-\frac{b B \left (a+b x^2\right )^{3/2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^8,x]

[Out]

-((b^2*B*Sqrt[a + b*x^2])/x) - (b*B*(a + b*x^2)^(3/2))/(3*x^3) - (B*(a + b*x^2)^
(5/2))/(5*x^5) - (A*(a + b*x^2)^(7/2))/(7*a*x^7) + b^(5/2)*B*ArcTanh[(Sqrt[b]*x)
/Sqrt[a + b*x^2]]

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Rubi in Sympy [A]  time = 17.5634, size = 95, normalized size = 0.88 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 a x^{7}} + B b^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )} - \frac{B b^{2} \sqrt{a + b x^{2}}}{x} - \frac{B b \left (a + b x^{2}\right )^{\frac{3}{2}}}{3 x^{3}} - \frac{B \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**8,x)

[Out]

-A*(a + b*x**2)**(7/2)/(7*a*x**7) + B*b**(5/2)*atanh(sqrt(b)*x/sqrt(a + b*x**2))
- B*b**2*sqrt(a + b*x**2)/x - B*b*(a + b*x**2)**(3/2)/(3*x**3) - B*(a + b*x**2)
**(5/2)/(5*x**5)

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Mathematica [A]  time = 0.168297, size = 106, normalized size = 0.98 \[ b^{5/2} B \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )-\frac{\sqrt{a+b x^2} \left (15 a^3 A+3 a^2 x^2 (7 a B+15 A b)+b^2 x^6 (161 a B+15 A b)+a b x^4 (77 a B+45 A b)\right )}{105 a x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^8,x]

[Out]

-(Sqrt[a + b*x^2]*(15*a^3*A + 3*a^2*(15*A*b + 7*a*B)*x^2 + a*b*(45*A*b + 77*a*B)
*x^4 + b^2*(15*A*b + 161*a*B)*x^6))/(105*a*x^7) + b^(5/2)*B*Log[b*x + Sqrt[b]*Sq
rt[a + b*x^2]]

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Maple [A]  time = 0.023, size = 155, normalized size = 1.4 \[ -{\frac{A}{7\,a{x}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{B}{5\,a{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Bb}{15\,{a}^{2}{x}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{8\,B{b}^{2}}{15\,{a}^{3}x} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{8\,B{b}^{3}x}{15\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{2\,B{b}^{3}x}{3\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{b}^{3}x}{a}\sqrt{b{x}^{2}+a}}+B{b}^{{\frac{5}{2}}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)*(B*x^2+A)/x^8,x)

[Out]

-1/7*A*(b*x^2+a)^(7/2)/a/x^7-1/5*B/a/x^5*(b*x^2+a)^(7/2)-2/15*B*b/a^2/x^3*(b*x^2
+a)^(7/2)-8/15*B*b^2/a^3/x*(b*x^2+a)^(7/2)+8/15*B*b^3/a^3*x*(b*x^2+a)^(5/2)+2/3*
B*b^3/a^2*x*(b*x^2+a)^(3/2)+B*b^3/a*x*(b*x^2+a)^(1/2)+B*b^(5/2)*ln(x*b^(1/2)+(b*
x^2+a)^(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)*(b*x^2 + a)^(5/2)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273926, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, B a b^{\frac{5}{2}} x^{7} \log \left (-2 \, b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left ({\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} x^{6} +{\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 3 \,{\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{210 \, a x^{7}}, \frac{105 \, B a \sqrt{-b} b^{2} x^{7} \arctan \left (\frac{b x}{\sqrt{b x^{2} + a} \sqrt{-b}}\right ) -{\left ({\left (161 \, B a b^{2} + 15 \, A b^{3}\right )} x^{6} +{\left (77 \, B a^{2} b + 45 \, A a b^{2}\right )} x^{4} + 15 \, A a^{3} + 3 \,{\left (7 \, B a^{3} + 15 \, A a^{2} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \, a x^{7}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^8,x, algorithm="fricas")

[Out]

[1/210*(105*B*a*b^(5/2)*x^7*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*
((161*B*a*b^2 + 15*A*b^3)*x^6 + (77*B*a^2*b + 45*A*a*b^2)*x^4 + 15*A*a^3 + 3*(7*
B*a^3 + 15*A*a^2*b)*x^2)*sqrt(b*x^2 + a))/(a*x^7), 1/105*(105*B*a*sqrt(-b)*b^2*x
^7*arctan(b*x/(sqrt(b*x^2 + a)*sqrt(-b))) - ((161*B*a*b^2 + 15*A*b^3)*x^6 + (77*
B*a^2*b + 45*A*a*b^2)*x^4 + 15*A*a^3 + 3*(7*B*a^3 + 15*A*a^2*b)*x^2)*sqrt(b*x^2
+ a))/(a*x^7)]

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Sympy [A]  time = 22.199, size = 592, normalized size = 5.48 \[ - \frac{15 A a^{7} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{33 A a^{6} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{17 A a^{5} b^{\frac{13}{2}} x^{4} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{3 A a^{4} b^{\frac{15}{2}} x^{6} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{12 A a^{3} b^{\frac{17}{2}} x^{8} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{8 A a^{2} b^{\frac{19}{2}} x^{10} \sqrt{\frac{a}{b x^{2}} + 1}}{105 a^{5} b^{4} x^{6} + 210 a^{4} b^{5} x^{8} + 105 a^{3} b^{6} x^{10}} - \frac{2 A a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{7 A b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 x^{2}} - \frac{A b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 a} - \frac{B \sqrt{a} b^{2}}{x \sqrt{1 + \frac{b x^{2}}{a}}} - \frac{B a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{5 x^{4}} - \frac{11 B a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15 x^{2}} - \frac{8 B b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{15} + B b^{\frac{5}{2}} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )} - \frac{B b^{3} x}{\sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**8,x)

[Out]

-15*A*a**7*b**(9/2)*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**
8 + 105*a**3*b**6*x**10) - 33*A*a**6*b**(11/2)*x**2*sqrt(a/(b*x**2) + 1)/(105*a*
*5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 17*A*a**5*b**(13/2)*x
**4*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**
6*x**10) - 3*A*a**4*b**(15/2)*x**6*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 21
0*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 12*A*a**3*b**(17/2)*x**8*sqrt(a/(b*x**
2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8 + 105*a**3*b**6*x**10) - 8*A*a*
*2*b**(19/2)*x**10*sqrt(a/(b*x**2) + 1)/(105*a**5*b**4*x**6 + 210*a**4*b**5*x**8
+ 105*a**3*b**6*x**10) - 2*A*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/(5*x**4) - 7*A*b**
(5/2)*sqrt(a/(b*x**2) + 1)/(15*x**2) - A*b**(7/2)*sqrt(a/(b*x**2) + 1)/(15*a) -
B*sqrt(a)*b**2/(x*sqrt(1 + b*x**2/a)) - B*a**2*sqrt(b)*sqrt(a/(b*x**2) + 1)/(5*x
**4) - 11*B*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/(15*x**2) - 8*B*b**(5/2)*sqrt(a/(b*x
**2) + 1)/15 + B*b**(5/2)*asinh(sqrt(b)*x/sqrt(a)) - B*b**3*x/(sqrt(a)*sqrt(1 +
b*x**2/a))

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GIAC/XCAS [A]  time = 0.245414, size = 432, normalized size = 4. \[ -\frac{1}{2} \, B b^{\frac{5}{2}}{\rm ln}\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2}\right ) + \frac{2 \,{\left (315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} B a b^{\frac{5}{2}} + 105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} A b^{\frac{7}{2}} - 1260 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} B a^{2} b^{\frac{5}{2}} + 2555 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a^{3} b^{\frac{5}{2}} + 525 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A a^{2} b^{\frac{7}{2}} - 3080 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{4} b^{\frac{5}{2}} + 2121 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{5} b^{\frac{5}{2}} + 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{4} b^{\frac{7}{2}} - 812 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{6} b^{\frac{5}{2}} + 161 \, B a^{7} b^{\frac{5}{2}} + 15 \, A a^{6} b^{\frac{7}{2}}\right )}}{105 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^8,x, algorithm="giac")

[Out]

-1/2*B*b^(5/2)*ln((sqrt(b)*x - sqrt(b*x^2 + a))^2) + 2/105*(315*(sqrt(b)*x - sqr
t(b*x^2 + a))^12*B*a*b^(5/2) + 105*(sqrt(b)*x - sqrt(b*x^2 + a))^12*A*b^(7/2) -
1260*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^2*b^(5/2) + 2555*(sqrt(b)*x - sqrt(b*x
^2 + a))^8*B*a^3*b^(5/2) + 525*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^2*b^(7/2) - 3
080*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^4*b^(5/2) + 2121*(sqrt(b)*x - sqrt(b*x^2
+ a))^4*B*a^5*b^(5/2) + 315*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^4*b^(7/2) - 812
*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^6*b^(5/2) + 161*B*a^7*b^(5/2) + 15*A*a^6*b^
(7/2))/((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^7