3.480 \(\int \frac{\sqrt{e x} (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=342 \[ \frac{e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{7/4} c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{\sqrt{e x} (a B+3 A c x)}{6 a^2 c \sqrt{a+c x^2}}-\frac{A e x \sqrt{a+c x^2}}{2 a^2 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt{e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

[Out]

-(Sqrt[e*x]*(a*B - A*c*x))/(3*a*c*(a + c*x^2)^(3/2)) + (Sqrt[e*x]*(a*B + 3*A*c*x
))/(6*a^2*c*Sqrt[a + c*x^2]) - (A*e*x*Sqrt[a + c*x^2])/(2*a^2*Sqrt[c]*Sqrt[e*x]*
(Sqrt[a] + Sqrt[c]*x)) + (A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sq
rt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^
(7/4)*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + ((Sqrt[a]*B - 3*A*Sqrt[c])*e*Sqrt[x]*
(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcT
an[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(12*a^(7/4)*c^(5/4)*Sqrt[e*x]*Sqrt[a + c*x^
2])

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Rubi [A]  time = 0.819793, antiderivative size = 342, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (\sqrt{a} B-3 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{7/4} c^{5/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{A e \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} c^{3/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{\sqrt{e x} (a B+3 A c x)}{6 a^2 c \sqrt{a+c x^2}}-\frac{A e x \sqrt{a+c x^2}}{2 a^2 \sqrt{c} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt{e x} (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[e*x]*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

-(Sqrt[e*x]*(a*B - A*c*x))/(3*a*c*(a + c*x^2)^(3/2)) + (Sqrt[e*x]*(a*B + 3*A*c*x
))/(6*a^2*c*Sqrt[a + c*x^2]) - (A*e*x*Sqrt[a + c*x^2])/(2*a^2*Sqrt[c]*Sqrt[e*x]*
(Sqrt[a] + Sqrt[c]*x)) + (A*e*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sq
rt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^
(7/4)*c^(3/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + ((Sqrt[a]*B - 3*A*Sqrt[c])*e*Sqrt[x]*
(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcT
an[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(12*a^(7/4)*c^(5/4)*Sqrt[e*x]*Sqrt[a + c*x^
2])

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Rubi in Sympy [A]  time = 105.636, size = 314, normalized size = 0.92 \[ - \frac{A e x \sqrt{a + c x^{2}}}{2 a^{2} \sqrt{c} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{A e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{7}{4}} c^{\frac{3}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{\sqrt{e x} \left (- A c x + B a\right )}{3 a c \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{e x} \left (\frac{3 A c x}{2} + \frac{B a}{2}\right )}{3 a^{2} c \sqrt{a + c x^{2}}} - \frac{e \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (3 A \sqrt{c} - B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{7}{4}} c^{\frac{5}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(1/2)*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

-A*e*x*sqrt(a + c*x**2)/(2*a**2*sqrt(c)*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) + A*e*s
qrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*ellipti
c_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(2*a**(7/4)*c**(3/4)*sqrt(e*x)*sqrt(
a + c*x**2)) - sqrt(e*x)*(-A*c*x + B*a)/(3*a*c*(a + c*x**2)**(3/2)) + sqrt(e*x)*
(3*A*c*x/2 + B*a/2)/(3*a**2*c*sqrt(a + c*x**2)) - e*sqrt(x)*sqrt((a + c*x**2)/(s
qrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(3*A*sqrt(c) - B*sqrt(a))*elliptic
_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(12*a**(7/4)*c**(5/4)*sqrt(e*x)*sqrt(
a + c*x**2))

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Mathematica [C]  time = 0.551874, size = 239, normalized size = 0.7 \[ \frac{e \left (-\sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a (3 A+B x)+c x^2 (A-B x)\right )-x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) \left (3 A \sqrt{c}-i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+3 A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{6 a^{3/2} c \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[e*x]*(A + B*x))/(a + c*x^2)^(5/2),x]

[Out]

(e*(-(Sqrt[a]*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(c*x^2*(A - B*x) + a*(3*A + B*x))) + 3*A
*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*(a + c*x^2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqr
t[a])/Sqrt[c]]/Sqrt[x]], -1] - ((-I)*Sqrt[a]*B + 3*A*Sqrt[c])*Sqrt[1 + a/(c*x^2)
]*x^(3/2)*(a + c*x^2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1
]))/(6*a^(3/2)*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c*Sqrt[e*x]*(a + c*x^2)^(3/2))

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Maple [A]  time = 0.022, size = 596, normalized size = 1.7 \[{\frac{1}{12\,{a}^{2}x{c}^{2}} \left ( 3\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}a{c}^{2}-6\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}a{c}^{2}+B\sqrt{-ac}\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}ac+3\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-6\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c+B\sqrt{-ac}\sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-ac}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-ac} \right ){\frac{1}{\sqrt{-ac}}}}},{\frac{\sqrt{2}}{2}} \right ){a}^{2}+6\,A{c}^{3}{x}^{4}+2\,aB{c}^{2}{x}^{3}+10\,aA{c}^{2}{x}^{2}-2\,{a}^{2}Bcx \right ) \sqrt{ex} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(1/2)*(B*x+A)/(c*x^2+a)^(5/2),x)

[Out]

1/12*(3*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))*x^2*a*c^2-6*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)*Ell
ipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*c^2+B*(-a*c)^(
1/2)*((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^2*a*c+3*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^2*c-6*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)*EllipticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))
*a^2*c+B*(-a*c)^(1/2)*((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^2+6*A*c^3*x^4+2*a*B*c^2*x^3+10*a*A*c^2
*x^2-2*a^2*B*c*x)*(e*x)^(1/2)/a^2/x/c^2/(c*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(e*x)/(c*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*sqrt(e*x)/(c*x^2 + a)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(e*x)/(c*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

integral((B*x + A)*sqrt(e*x)/((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)), 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((e*x)**(1/2)*(B*x+A)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} \sqrt{e x}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*sqrt(e*x)/(c*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*sqrt(e*x)/(c*x^2 + a)^(5/2), x)