3.516 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^7} \, dx\)

Optimal. Leaf size=120 \[ -\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac{\sqrt{a+b x^2} (A b-2 a B)}{8 a x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6} \]

[Out]

((A*b - 2*a*B)*Sqrt[a + b*x^2])/(8*a*x^4) + (b*(A*b - 2*a*B)*Sqrt[a + b*x^2])/(1
6*a^2*x^2) - (A*(a + b*x^2)^(3/2))/(6*a*x^6) - (b^2*(A*b - 2*a*B)*ArcTanh[Sqrt[a
+ b*x^2]/Sqrt[a]])/(16*a^(5/2))

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Rubi [A]  time = 0.253423, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{b^2 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 a^{5/2}}+\frac{b \sqrt{a+b x^2} (A b-2 a B)}{16 a^2 x^2}+\frac{\sqrt{a+b x^2} (A b-2 a B)}{8 a x^4}-\frac{A \left (a+b x^2\right )^{3/2}}{6 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - 2*a*B)*Sqrt[a + b*x^2])/(8*a*x^4) + (b*(A*b - 2*a*B)*Sqrt[a + b*x^2])/(1
6*a^2*x^2) - (A*(a + b*x^2)^(3/2))/(6*a*x^6) - (b^2*(A*b - 2*a*B)*ArcTanh[Sqrt[a
+ b*x^2]/Sqrt[a]])/(16*a^(5/2))

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Rubi in Sympy [A]  time = 21.0993, size = 107, normalized size = 0.89 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{3}{2}}}{6 a x^{6}} + \frac{\sqrt{a + b x^{2}} \left (\frac{A b}{2} - B a\right )}{4 a x^{4}} + \frac{b \sqrt{a + b x^{2}} \left (A b - 2 B a\right )}{16 a^{2} x^{2}} - \frac{b^{2} \left (\frac{A b}{2} - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*(a + b*x**2)**(3/2)/(6*a*x**6) + sqrt(a + b*x**2)*(A*b/2 - B*a)/(4*a*x**4) +
b*sqrt(a + b*x**2)*(A*b - 2*B*a)/(16*a**2*x**2) - b**2*(A*b/2 - B*a)*atanh(sqrt(
a + b*x**2)/sqrt(a))/(8*a**(5/2))

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Mathematica [A]  time = 0.165432, size = 123, normalized size = 1.02 \[ -\frac{b^2 (A b-2 a B) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{16 a^{5/2}}+\frac{b^2 \log (x) (A b-2 a B)}{16 a^{5/2}}+\sqrt{a+b x^2} \left (-\frac{b (2 a B-A b)}{16 a^2 x^2}+\frac{-6 a B-A b}{24 a x^4}-\frac{A}{6 x^6}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-A/(6*x^6) + (-(A*b) - 6*a*B)/(24*a*x^4) - (b*(-(A*b) + 2*a*B))/(16*a^2*x^2))*S
qrt[a + b*x^2] + (b^2*(A*b - 2*a*B)*Log[x])/(16*a^(5/2)) - (b^2*(A*b - 2*a*B)*Lo
g[a + Sqrt[a]*Sqrt[a + b*x^2]])/(16*a^(5/2))

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Maple [A]  time = 0.013, size = 197, normalized size = 1.6 \[ -{\frac{A}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ab}{8\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}A}{16\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{A{b}^{3}}{16\,{a}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{B}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{b}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B{b}^{2}}{8\,{a}^{2}}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*A*(b*x^2+a)^(3/2)/a/x^6+1/8*A*b/a^2/x^4*(b*x^2+a)^(3/2)-1/16*A*b^2/a^3/x^2*
(b*x^2+a)^(3/2)-1/16*A*b^3/a^(5/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+1/16*A*
b^3/a^3*(b*x^2+a)^(1/2)-1/4*B/a/x^4*(b*x^2+a)^(3/2)+1/8*B*b/a^2/x^2*(b*x^2+a)^(3
/2)+1/8*B*b^2/a^(3/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-1/8*B*b^2/a^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)*sqrt(b*x^2 + a)/x^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247209, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{6} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (3 \,{\left (2 \, B a b - A b^{2}\right )} x^{4} + 8 \, A a^{2} + 2 \,{\left (6 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{96 \, a^{\frac{5}{2}} x^{6}}, \frac{3 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (3 \,{\left (2 \, B a b - A b^{2}\right )} x^{4} + 8 \, A a^{2} + 2 \,{\left (6 \, B a^{2} + A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{48 \, \sqrt{-a} a^{2} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^7,x, algorithm="fricas")

[Out]

[-1/96*(3*(2*B*a*b^2 - A*b^3)*x^6*log(-((b*x^2 + 2*a)*sqrt(a) - 2*sqrt(b*x^2 + a
)*a)/x^2) + 2*(3*(2*B*a*b - A*b^2)*x^4 + 8*A*a^2 + 2*(6*B*a^2 + A*a*b)*x^2)*sqrt
(b*x^2 + a)*sqrt(a))/(a^(5/2)*x^6), 1/48*(3*(2*B*a*b^2 - A*b^3)*x^6*arctan(sqrt(
-a)/sqrt(b*x^2 + a)) - (3*(2*B*a*b - A*b^2)*x^4 + 8*A*a^2 + 2*(6*B*a^2 + A*a*b)*
x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^6)]

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Sympy [A]  time = 101.49, size = 226, normalized size = 1.88 \[ - \frac{A a}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{3}{2}}}{48 a x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{A b^{\frac{5}{2}}}{16 a^{2} x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 a^{\frac{5}{2}}} - \frac{B a}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B b^{\frac{3}{2}}}{8 a x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 5*A*sqrt(b)/(24*x**5*sqrt(a/(b*x**2
) + 1)) + A*b**(3/2)/(48*a*x**3*sqrt(a/(b*x**2) + 1)) + A*b**(5/2)/(16*a**2*x*sq
rt(a/(b*x**2) + 1)) - A*b**3*asinh(sqrt(a)/(sqrt(b)*x))/(16*a**(5/2)) - B*a/(4*s
qrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*B*sqrt(b)/(8*x**3*sqrt(a/(b*x**2) + 1)) -
B*b**(3/2)/(8*a*x*sqrt(a/(b*x**2) + 1)) + B*b**2*asinh(sqrt(a)/(sqrt(b)*x))/(8*a
**(3/2))

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GIAC/XCAS [A]  time = 0.229216, size = 189, normalized size = 1.58 \[ -\frac{\frac{3 \,{\left (2 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{6 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{3} - 6 \, \sqrt{b x^{2} + a} B a^{3} b^{3} - 3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{4} + 8 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{4} + 3 \, \sqrt{b x^{2} + a} A a^{2} b^{4}}{a^{2} b^{3} x^{6}}}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^7,x, algorithm="giac")

[Out]

-1/48*(3*(2*B*a*b^3 - A*b^4)*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (
6*(b*x^2 + a)^(5/2)*B*a*b^3 - 6*sqrt(b*x^2 + a)*B*a^3*b^3 - 3*(b*x^2 + a)^(5/2)*
A*b^4 + 8*(b*x^2 + a)^(3/2)*A*a*b^4 + 3*sqrt(b*x^2 + a)*A*a^2*b^4)/(a^2*b^3*x^6)
)/b