3.550 \(\int \frac{\left (a+b x^2\right )^{5/2} \left (A+B x^2\right )}{x^7} \, dx\)

Optimal. Leaf size=149 \[ \frac{5 b^2 \sqrt{a+b x^2} (6 a B+A b)}{16 a}-\frac{5 b^2 (6 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{5 b \left (a+b x^2\right )^{3/2} (6 a B+A b)}{48 a x^2}-\frac{\left (a+b x^2\right )^{5/2} (6 a B+A b)}{24 a x^4}-\frac{A \left (a+b x^2\right )^{7/2}}{6 a x^6} \]

[Out]

(5*b^2*(A*b + 6*a*B)*Sqrt[a + b*x^2])/(16*a) - (5*b*(A*b + 6*a*B)*(a + b*x^2)^(3
/2))/(48*a*x^2) - ((A*b + 6*a*B)*(a + b*x^2)^(5/2))/(24*a*x^4) - (A*(a + b*x^2)^
(7/2))/(6*a*x^6) - (5*b^2*(A*b + 6*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*Sq
rt[a])

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Rubi [A]  time = 0.28421, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{5 b^2 \sqrt{a+b x^2} (6 a B+A b)}{16 a}-\frac{5 b^2 (6 a B+A b) \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{16 \sqrt{a}}-\frac{5 b \left (a+b x^2\right )^{3/2} (6 a B+A b)}{48 a x^2}-\frac{\left (a+b x^2\right )^{5/2} (6 a B+A b)}{24 a x^4}-\frac{A \left (a+b x^2\right )^{7/2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(5/2)*(A + B*x^2))/x^7,x]

[Out]

(5*b^2*(A*b + 6*a*B)*Sqrt[a + b*x^2])/(16*a) - (5*b*(A*b + 6*a*B)*(a + b*x^2)^(3
/2))/(48*a*x^2) - ((A*b + 6*a*B)*(a + b*x^2)^(5/2))/(24*a*x^4) - (A*(a + b*x^2)^
(7/2))/(6*a*x^6) - (5*b^2*(A*b + 6*a*B)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*Sq
rt[a])

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Rubi in Sympy [A]  time = 24.3423, size = 136, normalized size = 0.91 \[ - \frac{A \left (a + b x^{2}\right )^{\frac{7}{2}}}{6 a x^{6}} + \frac{5 b^{2} \sqrt{a + b x^{2}} \left (A b + 6 B a\right )}{16 a} - \frac{5 b \left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b + 6 B a\right )}{48 a x^{2}} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \left (A b + 6 B a\right )}{24 a x^{4}} - \frac{5 b^{2} \left (A b + 6 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )}}{16 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**7,x)

[Out]

-A*(a + b*x**2)**(7/2)/(6*a*x**6) + 5*b**2*sqrt(a + b*x**2)*(A*b + 6*B*a)/(16*a)
- 5*b*(a + b*x**2)**(3/2)*(A*b + 6*B*a)/(48*a*x**2) - (a + b*x**2)**(5/2)*(A*b
+ 6*B*a)/(24*a*x**4) - 5*b**2*(A*b + 6*B*a)*atanh(sqrt(a + b*x**2)/sqrt(a))/(16*
sqrt(a))

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Mathematica [A]  time = 0.296847, size = 121, normalized size = 0.81 \[ \frac{1}{48} \left (\sqrt{a+b x^2} \left (-\frac{8 a^2 A}{x^6}-\frac{2 a (6 a B+13 A b)}{x^4}-\frac{3 b (18 a B+11 A b)}{x^2}+48 b^2 B\right )-\frac{15 b^2 (6 a B+A b) \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )}{\sqrt{a}}+\frac{15 b^2 \log (x) (6 a B+A b)}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(5/2)*(A + B*x^2))/x^7,x]

[Out]

((48*b^2*B - (8*a^2*A)/x^6 - (2*a*(13*A*b + 6*a*B))/x^4 - (3*b*(11*A*b + 18*a*B)
)/x^2)*Sqrt[a + b*x^2] + (15*b^2*(A*b + 6*a*B)*Log[x])/Sqrt[a] - (15*b^2*(A*b +
6*a*B)*Log[a + Sqrt[a]*Sqrt[a + b*x^2]])/Sqrt[a])/48

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Maple [B]  time = 0.013, size = 266, normalized size = 1.8 \[ -{\frac{A}{6\,a{x}^{6}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{Ab}{24\,{a}^{2}{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{b}^{2}A}{16\,{a}^{3}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{A{b}^{3}}{16\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,A{b}^{3}}{48\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{5\,A{b}^{3}}{16\,a}\sqrt{b{x}^{2}+a}}-{\frac{B}{4\,a{x}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{3\,Bb}{8\,{a}^{2}{x}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,B{b}^{2}}{8\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,B{b}^{2}}{8\,a} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,B{b}^{2}}{8}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) }+{\frac{15\,B{b}^{2}}{8}\sqrt{b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)*(B*x^2+A)/x^7,x)

[Out]

-1/6*A*(b*x^2+a)^(7/2)/a/x^6-1/24*A*b/a^2/x^4*(b*x^2+a)^(7/2)-1/16*A*b^2/a^3/x^2
*(b*x^2+a)^(7/2)+1/16*A*b^3/a^3*(b*x^2+a)^(5/2)+5/48*A*b^3/a^2*(b*x^2+a)^(3/2)-5
/16*A*b^3/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)+5/16*A*b^3/a*(b*x^2+a)^(
1/2)-1/4*B/a/x^4*(b*x^2+a)^(7/2)-3/8*B*b/a^2/x^2*(b*x^2+a)^(7/2)+3/8*B*b^2/a^2*(
b*x^2+a)^(5/2)+5/8*B*b^2/a*(b*x^2+a)^(3/2)-15/8*B*b^2*a^(1/2)*ln((2*a+2*a^(1/2)*
(b*x^2+a)^(1/2))/x)+15/8*B*b^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^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233035, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (6 \, B a b^{2} + A b^{3}\right )} x^{6} \log \left (-\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x^{2} + a} a}{x^{2}}\right ) + 2 \,{\left (48 \, B b^{2} x^{6} - 3 \,{\left (18 \, B a b + 11 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 2 \,{\left (6 \, B a^{2} + 13 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{a}}{96 \, \sqrt{a} x^{6}}, -\frac{15 \,{\left (6 \, B a b^{2} + A b^{3}\right )} x^{6} \arctan \left (\frac{\sqrt{-a}}{\sqrt{b x^{2} + a}}\right ) -{\left (48 \, B b^{2} x^{6} - 3 \,{\left (18 \, B a b + 11 \, A b^{2}\right )} x^{4} - 8 \, A a^{2} - 2 \,{\left (6 \, B a^{2} + 13 \, A a b\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-a}}{48 \, \sqrt{-a} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^7,x, algorithm="fricas")

[Out]

[1/96*(15*(6*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*(48*B*b^2*x^6 - 3*(18*B*a*b + 11*A*b^2)*x^4 - 8*A*a^2 - 2*(6*B*a^2
+ 13*A*a*b)*x^2)*sqrt(b*x^2 + a)*sqrt(a))/(sqrt(a)*x^6), -1/48*(15*(6*B*a*b^2 +
A*b^3)*x^6*arctan(sqrt(-a)/sqrt(b*x^2 + a)) - (48*B*b^2*x^6 - 3*(18*B*a*b + 11*
A*b^2)*x^4 - 8*A*a^2 - 2*(6*B*a^2 + 13*A*a*b)*x^2)*sqrt(b*x^2 + a)*sqrt(-a))/(sq
rt(-a)*x^6)]

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Sympy [A]  time = 163.34, size = 306, normalized size = 2.05 \[ - \frac{A a^{3}}{6 \sqrt{b} x^{7} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{17 A a^{2} \sqrt{b}}{24 x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{35 A a b^{\frac{3}{2}}}{48 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{A b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{2 x} - \frac{3 A b^{\frac{5}{2}}}{16 x \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{5 A b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{16 \sqrt{a}} - \frac{15 B \sqrt{a} b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{8} - \frac{B a^{3}}{4 \sqrt{b} x^{5} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{3 B a^{2} \sqrt{b}}{8 x^{3} \sqrt{\frac{a}{b x^{2}} + 1}} - \frac{B a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{x} + \frac{7 B a b^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{B b^{\frac{5}{2}} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(5/2)*(B*x**2+A)/x**7,x)

[Out]

-A*a**3/(6*sqrt(b)*x**7*sqrt(a/(b*x**2) + 1)) - 17*A*a**2*sqrt(b)/(24*x**5*sqrt(
a/(b*x**2) + 1)) - 35*A*a*b**(3/2)/(48*x**3*sqrt(a/(b*x**2) + 1)) - A*b**(5/2)*s
qrt(a/(b*x**2) + 1)/(2*x) - 3*A*b**(5/2)/(16*x*sqrt(a/(b*x**2) + 1)) - 5*A*b**3*
asinh(sqrt(a)/(sqrt(b)*x))/(16*sqrt(a)) - 15*B*sqrt(a)*b**2*asinh(sqrt(a)/(sqrt(
b)*x))/8 - B*a**3/(4*sqrt(b)*x**5*sqrt(a/(b*x**2) + 1)) - 3*B*a**2*sqrt(b)/(8*x*
*3*sqrt(a/(b*x**2) + 1)) - B*a*b**(3/2)*sqrt(a/(b*x**2) + 1)/x + 7*B*a*b**(3/2)/
(8*x*sqrt(a/(b*x**2) + 1)) + B*b**(5/2)*x/sqrt(a/(b*x**2) + 1)

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GIAC/XCAS [A]  time = 0.253609, size = 225, normalized size = 1.51 \[ \frac{48 \, \sqrt{b x^{2} + a} B b^{3} + \frac{15 \,{\left (6 \, B a b^{3} + A b^{4}\right )} \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{54 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} B a b^{3} - 96 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} B a^{2} b^{3} + 42 \, \sqrt{b x^{2} + a} B a^{3} b^{3} + 33 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} A b^{4} - 40 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a b^{4} + 15 \, \sqrt{b x^{2} + a} A a^{2} b^{4}}{b^{3} x^{6}}}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)/x^7,x, algorithm="giac")

[Out]

1/48*(48*sqrt(b*x^2 + a)*B*b^3 + 15*(6*B*a*b^3 + A*b^4)*arctan(sqrt(b*x^2 + a)/s
qrt(-a))/sqrt(-a) - (54*(b*x^2 + a)^(5/2)*B*a*b^3 - 96*(b*x^2 + a)^(3/2)*B*a^2*b
^3 + 42*sqrt(b*x^2 + a)*B*a^3*b^3 + 33*(b*x^2 + a)^(5/2)*A*b^4 - 40*(b*x^2 + a)^
(3/2)*A*a*b^4 + 15*sqrt(b*x^2 + a)*A*a^2*b^4)/(b^3*x^6))/b