3.791 \(\int \frac{\sqrt{a+b x^2} \left (A+B x^2\right )}{x^{9/2}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{2 b^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 a^{5/4} \sqrt{a+b x^2}}+\frac{2 \sqrt{a+b x^2} (A b-7 a B)}{21 a x^{3/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}} \]

[Out]

(2*(A*b - 7*a*B)*Sqrt[a + b*x^2])/(21*a*x^(3/2)) - (2*A*(a + b*x^2)^(3/2))/(7*a*
x^(7/2)) - (2*b^(3/4)*(A*b - 7*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])/(21*a^(
5/4)*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.229706, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{2 b^{3/4} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (A b-7 a B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 a^{5/4} \sqrt{a+b x^2}}+\frac{2 \sqrt{a+b x^2} (A b-7 a B)}{21 a x^{3/2}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{7 a x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(9/2),x]

[Out]

(2*(A*b - 7*a*B)*Sqrt[a + b*x^2])/(21*a*x^(3/2)) - (2*A*(a + b*x^2)^(3/2))/(7*a*
x^(7/2)) - (2*b^(3/4)*(A*b - 7*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])/(21*a^(
5/4)*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 20.3476, size = 139, normalized size = 0.91 \[ - \frac{2 A \left (a + b x^{2}\right )^{\frac{3}{2}}}{7 a x^{\frac{7}{2}}} + \frac{2 \sqrt{a + b x^{2}} \left (A b - 7 B a\right )}{21 a x^{\frac{3}{2}}} - \frac{2 b^{\frac{3}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b - 7 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{21 a^{\frac{5}{4}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*(b*x**2+a)**(1/2)/x**(9/2),x)

[Out]

-2*A*(a + b*x**2)**(3/2)/(7*a*x**(7/2)) + 2*sqrt(a + b*x**2)*(A*b - 7*B*a)/(21*a
*x**(3/2)) - 2*b**(3/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + s
qrt(b)*x)*(A*b - 7*B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(21*a
**(5/4)*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.247867, size = 139, normalized size = 0.91 \[ \sqrt{a+b x^2} \left (-\frac{2 (7 a B+2 A b)}{21 a x^{3/2}}-\frac{2 A}{7 x^{7/2}}\right )+\frac{4 i b x \sqrt{\frac{a}{b x^2}+1} (7 a B-A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{21 a \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[a + b*x^2]*(A + B*x^2))/x^(9/2),x]

[Out]

((-2*A)/(7*x^(7/2)) - (2*(2*A*b + 7*a*B))/(21*a*x^(3/2)))*Sqrt[a + b*x^2] + (((4
*I)/21)*b*(-(A*b) + 7*a*B)*Sqrt[1 + a/(b*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqr
t[a])/Sqrt[b]]/Sqrt[x]], -1])/(a*Sqrt[(I*Sqrt[a])/Sqrt[b]]*Sqrt[a + b*x^2])

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Maple [A]  time = 0.058, size = 242, normalized size = 1.6 \[ -{\frac{2}{21\,a} \left ( A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-ab}{x}^{3}b-7\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) \sqrt{-ab}{x}^{3}a+2\,A{b}^{2}{x}^{4}+7\,B{x}^{4}ab+5\,aAb{x}^{2}+7\,B{x}^{2}{a}^{2}+3\,A{a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{x}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*(b*x^2+a)^(1/2)/x^(9/2),x)

[Out]

-2/21/(b*x^2+a)^(1/2)*(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^3*b-7*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^3*a+2*A*b^2*x^4+7*B*x^4*a*b+5*a*A*b*x^2+7*B*x^2*a^2+3*A*a^2)/
x^(7/2)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2),x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{9}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2),x, algorithm="fricas")

[Out]

integral((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2), 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**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2),x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*sqrt(b*x^2 + a)/x^(9/2), x)