3.799 \(\int \frac{\left (a+b x^2\right )^{3/2} \left (A+B x^2\right )}{(e x)^{7/2}} \, dx\)

Optimal. Leaf size=365 \[ \frac{12 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}-\frac{24 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}+\frac{12 b (e x)^{3/2} \sqrt{a+b x^2} (a B+A b)}{5 a e^5}+\frac{24 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{5 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \left (a+b x^2\right )^{3/2} (a B+A b)}{a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}} \]

[Out]

(12*b*(A*b + a*B)*(e*x)^(3/2)*Sqrt[a + b*x^2])/(5*a*e^5) + (24*Sqrt[b]*(A*b + a*
B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*e^4*(Sqrt[a] + Sqrt[b]*x)) - (2*(A*b + a*B)*(a
+ b*x^2)^(3/2))/(a*e^3*Sqrt[e*x]) - (2*A*(a + b*x^2)^(5/2))/(5*a*e*(e*x)^(5/2))
- (24*a^(1/4)*b^(1/4)*(A*b + 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*e^(7/2)*Sqrt[a + b*x^2]) + (12*a^(1/4)*b^(1/4)*(A*b + a*B)*(Sqrt[a] + Sqrt
[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqr
t[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*e^(7/2)*Sqrt[a + b*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.701946, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{12 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}-\frac{24 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} (a B+A b) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}}\right )|\frac{1}{2}\right )}{5 e^{7/2} \sqrt{a+b x^2}}+\frac{12 b (e x)^{3/2} \sqrt{a+b x^2} (a B+A b)}{5 a e^5}+\frac{24 \sqrt{b} \sqrt{e x} \sqrt{a+b x^2} (a B+A b)}{5 e^4 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{2 \left (a+b x^2\right )^{3/2} (a B+A b)}{a e^3 \sqrt{e x}}-\frac{2 A \left (a+b x^2\right )^{5/2}}{5 a e (e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(7/2),x]

[Out]

(12*b*(A*b + a*B)*(e*x)^(3/2)*Sqrt[a + b*x^2])/(5*a*e^5) + (24*Sqrt[b]*(A*b + a*
B)*Sqrt[e*x]*Sqrt[a + b*x^2])/(5*e^4*(Sqrt[a] + Sqrt[b]*x)) - (2*(A*b + a*B)*(a
+ b*x^2)^(3/2))/(a*e^3*Sqrt[e*x]) - (2*A*(a + b*x^2)^(5/2))/(5*a*e*(e*x)^(5/2))
- (24*a^(1/4)*b^(1/4)*(A*b + 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*e^(7/2)*Sqrt[a + b*x^2]) + (12*a^(1/4)*b^(1/4)*(A*b + a*B)*(Sqrt[a] + Sqrt
[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*ArcTan[(b^(1/4)*Sqr
t[e*x])/(a^(1/4)*Sqrt[e])], 1/2])/(5*e^(7/2)*Sqrt[a + b*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 75.0598, size = 337, normalized size = 0.92 \[ - \frac{2 A \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 a e \left (e x\right )^{\frac{5}{2}}} - \frac{24 \sqrt [4]{a} \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 e^{\frac{7}{2}} \sqrt{a + b x^{2}}} + \frac{12 \sqrt [4]{a} \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \left (A b + B a\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt [4]{a} \sqrt{e}} \right )}\middle | \frac{1}{2}\right )}{5 e^{\frac{7}{2}} \sqrt{a + b x^{2}}} + \frac{24 \sqrt{b} \sqrt{e x} \sqrt{a + b x^{2}} \left (A b + B a\right )}{5 e^{4} \left (\sqrt{a} + \sqrt{b} x\right )} + \frac{12 b \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (A b + B a\right )}{5 a e^{5}} - \frac{2 \left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b + B a\right )}{a e^{3} \sqrt{e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(3/2)*(B*x**2+A)/(e*x)**(7/2),x)

[Out]

-2*A*(a + b*x**2)**(5/2)/(5*a*e*(e*x)**(5/2)) - 24*a**(1/4)*b**(1/4)*sqrt((a + b
*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*(A*b + B*a)*elliptic_e(2*
atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1/2)/(5*e**(7/2)*sqrt(a + b*x**2))
+ 12*a**(1/4)*b**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sq
rt(b)*x)*(A*b + B*a)*elliptic_f(2*atan(b**(1/4)*sqrt(e*x)/(a**(1/4)*sqrt(e))), 1
/2)/(5*e**(7/2)*sqrt(a + b*x**2)) + 24*sqrt(b)*sqrt(e*x)*sqrt(a + b*x**2)*(A*b +
B*a)/(5*e**4*(sqrt(a) + sqrt(b)*x)) + 12*b*(e*x)**(3/2)*sqrt(a + b*x**2)*(A*b +
B*a)/(5*a*e**5) - 2*(a + b*x**2)**(3/2)*(A*b + B*a)/(a*e**3*sqrt(e*x))

_______________________________________________________________________________________

Mathematica [C]  time = 1.02817, size = 232, normalized size = 0.64 \[ \frac{x \left (24 \sqrt{a} \sqrt{b} x^{7/2} \sqrt{\frac{a}{b x^2}+1} (a B+A b) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )-24 \sqrt{a} \sqrt{b} x^{7/2} \sqrt{\frac{a}{b x^2}+1} (a B+A b) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )+2 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} \left (a+b x^2\right ) \left (-a A+7 a B x^2+5 A b x^2+b B x^4\right )\right )}{5 \sqrt{\frac{i \sqrt{a}}{\sqrt{b}}} (e x)^{7/2} \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^(3/2)*(A + B*x^2))/(e*x)^(7/2),x]

[Out]

(x*(2*Sqrt[(I*Sqrt[a])/Sqrt[b]]*(a + b*x^2)*(-(a*A) + 5*A*b*x^2 + 7*a*B*x^2 + b*
B*x^4) - 24*Sqrt[a]*Sqrt[b]*(A*b + a*B)*Sqrt[1 + a/(b*x^2)]*x^(7/2)*EllipticE[I*
ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[x]], -1] + 24*Sqrt[a]*Sqrt[b]*(A*b + a*B)
*Sqrt[1 + a/(b*x^2)]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[b]]/Sqrt[
x]], -1]))/(5*Sqrt[(I*Sqrt[a])/Sqrt[b]]*(e*x)^(7/2)*Sqrt[a + b*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.027, size = 422, normalized size = 1.2 \[{\frac{2}{5\,{x}^{2}{e}^{3}} \left ( 12\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab-6\,A\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}ab+12\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}-6\,B\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}+B{b}^{2}{x}^{6}-7\,A{b}^{2}{x}^{4}-4\,B{x}^{4}ab-8\,aAb{x}^{2}-5\,B{x}^{2}{a}^{2}-A{a}^{2} \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(B*x^2+A)/(e*x)^(7/2),x)

[Out]

2/5/x^2*(12*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-6*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)*E
llipticF(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b+12*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^2-6*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+B*b^2*x^6-7*A*b^2*x^4-4*B*x^4
*a*b-8*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)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )}{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/(e*x)^(7/2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B b x^{4} +{\left (B a + A b\right )} x^{2} + A a\right )} \sqrt{b x^{2} + a}}{\sqrt{e x} e^{3} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/(e*x)^(7/2),x, algorithm="fricas")

[Out]

integral((B*b*x^4 + (B*a + A*b)*x^2 + A*a)*sqrt(b*x^2 + a)/(sqrt(e*x)*e^3*x^3),
x)

_______________________________________________________________________________________

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)**(3/2)*(B*x**2+A)/(e*x)**(7/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )}{\left (b x^{2} + a\right )}^{\frac{3}{2}}}{\left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(3/2)/(e*x)^(7/2),x, algorithm="giac")

[Out]

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