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

Optimal. Leaf size=338 \[ \frac{2 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt{a+b x^2}}-\frac{4 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt{a+b x^2}}+\frac{4 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (5 a B+A b)}{5 a e^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \sqrt{a+b x^2} (5 a B+A b)}{5 a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}} \]

[Out]

(-2*(A*b + 5*a*B)*Sqrt[a + b*x^2])/(5*a*e^3*Sqrt[e*x]) + (4*Sqrt[b]*(A*b + 5*a*B
)*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*a*e^4*(Sqrt[a] + Sqrt[b]*x)) - (2*A*(a + b*x^2)^
(3/2))/(5*a*e*(e*x)^(5/2)) - (4*b^(1/4)*(A*b + 5*a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt
[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^
(1/4)*Sqrt[e])], 1/2])/(5*a^(3/4)*e^(7/2)*Sqrt[a + b*x^2]) + (2*b^(1/4)*(A*b + 5
*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[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*a^(3/4)*e^(7/2)*Sqrt[a
+ b*x^2])

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Rubi [A]  time = 0.646433, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt{a+b x^2}}-\frac{4 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (5 a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^{7/2} \sqrt{a+b x^2}}+\frac{4 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (5 a B+A b)}{5 a e^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \sqrt{a+b x^2} (5 a B+A b)}{5 a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{3/2}}{5 a e (e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(A*b + 5*a*B)*Sqrt[a + b*x^2])/(5*a*e^3*Sqrt[e*x]) + (4*Sqrt[b]*(A*b + 5*a*B
)*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*a*e^4*(Sqrt[a] + Sqrt[b]*x)) - (2*A*(a + b*x^2)^
(3/2))/(5*a*e*(e*x)^(5/2)) - (4*b^(1/4)*(A*b + 5*a*B)*(Sqrt[a] + Sqrt[b]*x)*Sqrt
[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/4)*Sqrt[e*x])/(a^
(1/4)*Sqrt[e])], 1/2])/(5*a^(3/4)*e^(7/2)*Sqrt[a + b*x^2]) + (2*b^(1/4)*(A*b + 5
*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[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*a^(3/4)*e^(7/2)*Sqrt[a
+ b*x^2])

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Rubi in Sympy [A]  time = 65.6255, size = 313, normalized size = 0.93 \[ - \frac{2 A \left (a + b x^{2}\right )^{\frac{3}{2}}}{5 a e \left (e x\right )^{\frac{5}{2}}} + \frac{4 \sqrt{b} \sqrt{e x} \sqrt{a + b x^{2}} \left (A b + 5 B a\right )}{5 a e^{4} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{2 \sqrt{a + b x^{2}} \left (A b + 5 B a\right )}{5 a e^{3} \sqrt{e x}} - \frac{4 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + 5 B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{a + b x^{2}}} + \frac{2 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + 5 B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{3}{4}} e^{\frac{7}{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*(a + b*x**2)**(3/2)/(5*a*e*(e*x)**(5/2)) + 4*sqrt(b)*sqrt(e*x)*sqrt(a + b*x
**2)*(A*b + 5*B*a)/(5*a*e**4*(sqrt(a) + sqrt(b)*x)) - 2*sqrt(a + b*x**2)*(A*b +
5*B*a)/(5*a*e**3*sqrt(e*x)) - 4*b**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)
**2)*(sqrt(a) + sqrt(b)*x)*(A*b + 5*B*a)*elliptic_e(2*atan(b**(1/4)*sqrt(e*x)/(a
**(1/4)*sqrt(e))), 1/2)/(5*a**(3/4)*e**(7/2)*sqrt(a + b*x**2)) + 2*b**(1/4)*sqrt
((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(A*b + 5*B*a)*elli
ptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(5*a**(3/4)*e**(7/2)*
sqrt(a + b*x**2))

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Mathematica [C]  time = 0.895357, size = 217, normalized size = 0.64 \[ \frac{x \left (-2 \sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right ) \left (A-5 B x^2\right )+4 \sqrt{b} x^{7/2} \sqrt{\frac{a}{b x^2}+1} (5 a B+A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-4 \sqrt{b} x^{7/2} \sqrt{\frac{a}{b x^2}+1} (5 a B+A b) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{5 \sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} (e x)^{7/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-2*Sqrt[a]*Sqrt[(I*Sqrt[a])/Sqrt[b]]*(a + b*x^2)*(A - 5*B*x^2) - 4*Sqrt[b]*(
A*b + 5*a*B)*Sqrt[1 + a/(b*x^2)]*x^(7/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sq
rt[b]]/Sqrt[x]], -1] + 4*Sqrt[b]*(A*b + 5*a*B)*Sqrt[1 + a/(b*x^2)]*x^(7/2)*Ellip
ticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1]))/(5*Sqrt[a]*Sqrt[(I*Sqrt
[a])/Sqrt[b]]*(e*x)^(7/2)*Sqrt[a + b*x^2])

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Maple [A]  time = 0.049, size = 417, normalized size = 1.2 \[{\frac{2}{5\,{x}^{2}{e}^{3}a} \left ( 2\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-A\sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{2}\sqrt{{1 \left ( -bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( bx+\sqrt{-ab} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}ab+10\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}-5\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}-2\,A{b}^{2}{x}^{4}-5\,B{x}^{4}ab-3\,aAb{x}^{2}-5\,B{x}^{2}{a}^{2}-A{a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5/x^2*(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)*EllipticE(((b*x+(-a*b)^(1/2))/(-
a*b)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b-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)*Elli
pticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b+10*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)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2
*2^(1/2))*x^2*a^2-5*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))*x^2*a^2-2*A*b^2*x^4-5*B*x^4*a*b-3*a*A*b*
x^2-5*B*x^2*a^2-A*a^2)/(b*x^2+a)^(1/2)/e^3/(e*x)^(1/2)/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*sqrt(b*x^2 + a)/(e*x)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{\sqrt{e x} e^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x^2 + A)*sqrt(b*x^2 + a)/(sqrt(e*x)*e^3*x^3), 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)/(e*x)**(7/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} \sqrt{b x^{2} + a}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x^2 + A)*sqrt(b*x^2 + a)/(e*x)^(7/2), x)