3.441 \(\int \frac{(A+B x) \sqrt{a+c x^2}}{(e x)^{9/2}} \, dx\)

Optimal. Leaf size=368 \[ \frac{2 c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{5/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}+\frac{4 B c^{3/2} x \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}} \]

[Out]

(-4*A*c*Sqrt[a + c*x^2])/(21*a*e^3*(e*x)^(3/2)) - (4*B*c*Sqrt[a + c*x^2])/(5*a*e
^4*Sqrt[e*x]) - (2*(5*A + 7*B*x)*Sqrt[a + c*x^2])/(35*e*(e*x)^(7/2)) + (4*B*c^(3
/2)*x*Sqrt[a + c*x^2])/(5*a*e^4*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*B*c^(5/4)*
Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elliptic
E[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*a^(3/4)*e^4*Sqrt[e*x]*Sqrt[a + c
*x^2]) + (2*(21*Sqrt[a]*B - 5*A*Sqrt[c])*c^(5/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*S
qrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^
(1/4)], 1/2])/(105*a^(5/4)*e^4*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.994203, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{2 c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (21 \sqrt{a} B-5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 a^{5/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{4 B c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} e^4 \sqrt{e x} \sqrt{a+c x^2}}-\frac{2 \sqrt{a+c x^2} (5 A+7 B x)}{35 e (e x)^{7/2}}-\frac{4 A c \sqrt{a+c x^2}}{21 a e^3 (e x)^{3/2}}+\frac{4 B c^{3/2} x \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{4 B c \sqrt{a+c x^2}}{5 a e^4 \sqrt{e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*A*c*Sqrt[a + c*x^2])/(21*a*e^3*(e*x)^(3/2)) - (4*B*c*Sqrt[a + c*x^2])/(5*a*e
^4*Sqrt[e*x]) - (2*(5*A + 7*B*x)*Sqrt[a + c*x^2])/(35*e*(e*x)^(7/2)) + (4*B*c^(3
/2)*x*Sqrt[a + c*x^2])/(5*a*e^4*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*B*c^(5/4)*
Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elliptic
E[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*a^(3/4)*e^4*Sqrt[e*x]*Sqrt[a + c
*x^2]) + (2*(21*Sqrt[a]*B - 5*A*Sqrt[c])*c^(5/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*S
qrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^
(1/4)], 1/2])/(105*a^(5/4)*e^4*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 137.94, size = 348, normalized size = 0.95 \[ - \frac{4 A c \sqrt{a + c x^{2}}}{21 a e^{3} \left (e x\right )^{\frac{3}{2}}} + \frac{4 B c^{\frac{3}{2}} x \sqrt{a + c x^{2}}}{5 a e^{4} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{4 B c \sqrt{a + c x^{2}}}{5 a e^{4} \sqrt{e x}} - \frac{4 B c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{3}{4}} e^{4} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{4 \left (\frac{5 A}{2} + \frac{7 B x}{2}\right ) \sqrt{a + c x^{2}}}{35 e \left (e x\right )^{\frac{7}{2}}} - \frac{2 c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (5 A \sqrt{c} - 21 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 a^{\frac{5}{4}} e^{4} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*A*c*sqrt(a + c*x**2)/(21*a*e**3*(e*x)**(3/2)) + 4*B*c**(3/2)*x*sqrt(a + c*x**
2)/(5*a*e**4*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) - 4*B*c*sqrt(a + c*x**2)/(5*a*e**4
*sqrt(e*x)) - 4*B*c**(5/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(
sqrt(a) + sqrt(c)*x)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(5*a**(3
/4)*e**4*sqrt(e*x)*sqrt(a + c*x**2)) - 4*(5*A/2 + 7*B*x/2)*sqrt(a + c*x**2)/(35*
e*(e*x)**(7/2)) - 2*c**(5/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)
*(sqrt(a) + sqrt(c)*x)*(5*A*sqrt(c) - 21*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)*s
qrt(x)/a**(1/4)), 1/2)/(105*a**(5/4)*e**4*sqrt(e*x)*sqrt(a + c*x**2))

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Mathematica [C]  time = 1.15194, size = 236, normalized size = 0.64 \[ -\frac{2 \sqrt{e x} \left (2 i c^{3/2} x^{9/2} \sqrt{\frac{a}{c x^2}+1} \left (5 A \sqrt{c}+21 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (3 a (5 A+7 B x)+10 A c x^2\right )+42 \sqrt{a} B c^{3/2} x^{9/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{105 a e^5 x^4 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[e*x]*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(10*A*c*x^2 + 3*a*(5*A + 7*
B*x)) + 42*Sqrt[a]*B*c^(3/2)*Sqrt[1 + a/(c*x^2)]*x^(9/2)*EllipticE[I*ArcSinh[Sqr
t[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + (2*I)*((21*I)*Sqrt[a]*B + 5*A*Sqrt[c])*c^
(3/2)*Sqrt[1 + a/(c*x^2)]*x^(9/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/
Sqrt[x]], -1]))/(105*a*Sqrt[(I*Sqrt[a])/Sqrt[c]]*e^5*x^4*Sqrt[a + c*x^2])

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Maple [A]  time = 0.048, size = 340, normalized size = 0.9 \[ -{\frac{2}{105\,{x}^{3}{e}^{4}a} \left ( 5\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{3}c+21\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}ac-42\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{3}ac+42\,B{c}^{2}{x}^{5}+10\,A{c}^{2}{x}^{4}+63\,aBc{x}^{3}+25\,aAc{x}^{2}+21\,{a}^{2}Bx+15\,A{a}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105/x^3*(5*A*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1
/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))
/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*(-a*c)^(1/2)*x^3*c+21*B*((c*x+(-a*c)^(1/2))/(-
a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^
(1/2))^(1/2)*EllipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^3*
a*c-42*B*((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-a*c)^(1/2))/(-
a*c)^(1/2))^(1/2)*(-x*c/(-a*c)^(1/2))^(1/2)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)
^(1/2))^(1/2),1/2*2^(1/2))*x^3*a*c+42*B*c^2*x^5+10*A*c^2*x^4+63*a*B*c*x^3+25*a*A
*c*x^2+21*a^2*B*x+15*A*a^2)/(c*x^2+a)^(1/2)/e^4/(e*x)^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^2 + a)*(B*x + A)/(sqrt(e*x)*e^4*x^4), 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+A)*(c*x**2+a)**(1/2)/(e*x)**(9/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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