3.793 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^{13/2}} \, dx\)

Optimal. Leaf size=187 \[ \frac{2 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-11 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 a^{9/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (5 A b-11 a B)}{231 a^2 x^{3/2}}+\frac{2 \sqrt{a+b x^2} (5 A b-11 a B)}{77 a x^{7/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}} \]

[Out]

(2*(5*A*b - 11*a*B)*Sqrt[a + b*x^2])/(77*a*x^(7/2)) + (4*b*(5*A*b - 11*a*B)*Sqrt
[a + b*x^2])/(231*a^2*x^(3/2)) - (2*A*(a + b*x^2)^(3/2))/(11*a*x^(11/2)) + (2*b^
(7/4)*(5*A*b - 11*a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]
*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(231*a^(9/4)*Sqrt[a
+ b*x^2])

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Rubi [A]  time = 0.278419, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 b^{7/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 A b-11 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{231 a^{9/4} \sqrt{a+b x^2}}+\frac{4 b \sqrt{a+b x^2} (5 A b-11 a B)}{231 a^2 x^{3/2}}+\frac{2 \sqrt{a+b x^2} (5 A b-11 a B)}{77 a x^{7/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{11 a x^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(13/2),x]

[Out]

(2*(5*A*b - 11*a*B)*Sqrt[a + b*x^2])/(77*a*x^(7/2)) + (4*b*(5*A*b - 11*a*B)*Sqrt
[a + b*x^2])/(231*a^2*x^(3/2)) - (2*A*(a + b*x^2)^(3/2))/(11*a*x^(11/2)) + (2*b^
(7/4)*(5*A*b - 11*a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]
*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(231*a^(9/4)*Sqrt[a
+ b*x^2])

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Rubi in Sympy [A]  time = 25.4306, size = 177, normalized size = 0.95 \[ - \frac{2 A \left (a + b x^{2}\right )^{\frac{3}{2}}}{11 a x^{\frac{11}{2}}} + \frac{2 \sqrt{a + b x^{2}} \left (5 A b - 11 B a\right )}{77 a x^{\frac{7}{2}}} + \frac{4 b \sqrt{a + b x^{2}} \left (5 A b - 11 B a\right )}{231 a^{2} x^{\frac{3}{2}}} + \frac{2 b^{\frac{7}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (5 A b - 11 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{231 a^{\frac{9}{4}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**(13/2),x)

[Out]

-2*A*(a + b*x**2)**(3/2)/(11*a*x**(11/2)) + 2*sqrt(a + b*x**2)*(5*A*b - 11*B*a)/
(77*a*x**(7/2)) + 4*b*sqrt(a + b*x**2)*(5*A*b - 11*B*a)/(231*a**2*x**(3/2)) + 2*
b**(7/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(5*A*
b - 11*B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(231*a**(9/4)*sqr
t(a + b*x**2))

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Mathematica [C]  time = 0.319543, size = 163, normalized size = 0.87 \[ \sqrt{a+b x^2} \left (-\frac{4 b (11 a B-5 A b)}{231 a^2 x^{3/2}}-\frac{2 (11 a B+2 A b)}{77 a x^{7/2}}-\frac{2 A}{11 x^{11/2}}\right )-\frac{4 i b^2 x \sqrt{\frac{a}{b x^2}+1} (11 a B-5 A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{231 a^2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(13/2),x]

[Out]

((-2*A)/(11*x^(11/2)) - (2*(2*A*b + 11*a*B))/(77*a*x^(7/2)) - (4*b*(-5*A*b + 11*
a*B))/(231*a^2*x^(3/2)))*Sqrt[a + b*x^2] - (((4*I)/231)*b^2*(-5*A*b + 11*a*B)*Sq
rt[1 + a/(b*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1])
/(a^2*Sqrt[(I*Sqrt[a])/Sqrt[b]]*Sqrt[a + b*x^2])

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Maple [A]  time = 0.046, size = 270, normalized size = 1.4 \[{\frac{2}{231\,{a}^{2}} \left ( 5\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{5}{b}^{2}-11\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{5}ab+10\,A{x}^{6}{b}^{3}-22\,B{x}^{6}a{b}^{2}+4\,A{x}^{4}a{b}^{2}-55\,B{x}^{4}{a}^{2}b-27\,A{x}^{2}{a}^{2}b-33\,B{x}^{2}{a}^{3}-21\,A{a}^{3} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{x}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(b*x^2+a)^(1/2)/x^(13/2),x)

[Out]

2/231/(b*x^2+a)^(1/2)*(5*A*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*
x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-
a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*x^5*b^2-11*B*((b*x+(-a
*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*
(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*
2^(1/2))*(-a*b)^(1/2)*x^5*a*b+10*A*x^6*b^3-22*B*x^6*a*b^2+4*A*x^4*a*b^2-55*B*x^4
*a^2*b-27*A*x^2*a^2*b-33*B*x^2*a^3-21*A*a^3)/x^(11/2)/a^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{13}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2),x, algorithm="fricas")

[Out]

integral((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(13/2), x)