3.482 \(\int \frac{A+B x}{(e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=373 \[ \frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{11/4} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}+\frac{7 A \sqrt{c} x \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}} \]

[Out]

(A + B*x)/(3*a*e*Sqrt[e*x]*(a + c*x^2)^(3/2)) + (7*A + 5*B*x)/(6*a^2*e*Sqrt[e*x]
*Sqrt[a + c*x^2]) - (7*A*Sqrt[a + c*x^2])/(2*a^3*e*Sqrt[e*x]) + (7*A*Sqrt[c]*x*S
qrt[a + c*x^2])/(2*a^3*e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (7*A*c^(1/4)*Sqrt[x]
*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*Arc
Tan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(11/4)*e*Sqrt[e*x]*Sqrt[a + c*x^2]) +
((5*Sqrt[a]*B + 21*A*Sqrt[c])*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(S
qrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(12*
a^(11/4)*c^(1/4)*e*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.01616, antiderivative size = 373, 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 \[ \frac{\sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+21 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 a^{11/4} \sqrt [4]{c} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{11/4} e \sqrt{e x} \sqrt{a+c x^2}}-\frac{7 A \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x}}+\frac{7 A \sqrt{c} x \sqrt{a+c x^2}}{2 a^3 e \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{7 A+5 B x}{6 a^2 e \sqrt{e x} \sqrt{a+c x^2}}+\frac{A+B x}{3 a e \sqrt{e x} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(A + B*x)/(3*a*e*Sqrt[e*x]*(a + c*x^2)^(3/2)) + (7*A + 5*B*x)/(6*a^2*e*Sqrt[e*x]
*Sqrt[a + c*x^2]) - (7*A*Sqrt[a + c*x^2])/(2*a^3*e*Sqrt[e*x]) + (7*A*Sqrt[c]*x*S
qrt[a + c*x^2])/(2*a^3*e*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (7*A*c^(1/4)*Sqrt[x]
*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*Arc
Tan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(2*a^(11/4)*e*Sqrt[e*x]*Sqrt[a + c*x^2]) +
((5*Sqrt[a]*B + 21*A*Sqrt[c])*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(S
qrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(12*
a^(11/4)*c^(1/4)*e*Sqrt[e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 148.684, size = 342, normalized size = 0.92 \[ \frac{7 A \sqrt{c} x \sqrt{a + c x^{2}}}{2 a^{3} e \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{7 A \sqrt{a + c x^{2}}}{2 a^{3} e \sqrt{e x}} - \frac{7 A \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{11}{4}} e \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{A + B x}{3 a e \sqrt{e x} \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{\frac{7 A}{2} + \frac{5 B x}{2}}{3 a^{2} e \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{\sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (21 A \sqrt{c} + 5 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{12 a^{\frac{11}{4}} \sqrt [4]{c} e \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x)**(3/2)/(c*x**2+a)**(5/2),x)

[Out]

7*A*sqrt(c)*x*sqrt(a + c*x**2)/(2*a**3*e*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) - 7*A*
sqrt(a + c*x**2)/(2*a**3*e*sqrt(e*x)) - 7*A*c**(1/4)*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)/(2*a**(11/4)*e*sqrt(e*x)*sqrt(a + c*x**2)) + (A + B*x)/(3*a*e*
sqrt(e*x)*(a + c*x**2)**(3/2)) + (7*A/2 + 5*B*x/2)/(3*a**2*e*sqrt(e*x)*sqrt(a +
c*x**2)) + sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c
)*x)*(21*A*sqrt(c) + 5*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)),
1/2)/(12*a**(11/4)*c**(1/4)*e*sqrt(e*x)*sqrt(a + c*x**2))

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Mathematica [C]  time = 0.903615, size = 237, normalized size = 0.64 \[ \frac{x \left (\sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a (9 A+7 B x)+c x^2 (7 A+5 B x)\right )+x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) \left (21 A \sqrt{c}+5 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-21 A \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{6 a^{5/2} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{3/2} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(x*(Sqrt[a]*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(c*x^2*(7*A + 5*B*x) + a*(9*A + 7*B*x)) -
21*A*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*(a + c*x^2)*EllipticE[I*ArcSinh[Sqrt[(I
*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + ((5*I)*Sqrt[a]*B + 21*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^(5/2)*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(e*x)^(3/2)*(a + c*x^2)^(3/2))

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Maple [A]  time = 0.034, size = 605, normalized size = 1.6 \[{\frac{1}{12\,{a}^{3}ec} \left ( 42\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}a{c}^{2}-21\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}a{c}^{2}+5\,B\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac+42\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c-21\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}c+5\,B\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{2}-42\,A{c}^{3}{x}^{4}+10\,aB{c}^{2}{x}^{3}-70\,aA{c}^{2}{x}^{2}+14\,{a}^{2}Bcx-24\,A{a}^{2}c \right ){\frac{1}{\sqrt{ex}}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x)^(3/2)/(c*x^2+a)^(5/2),x)

[Out]

1/12*(42*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))*x^2*a*c^2-21*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)*E
llipticF(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*c^2+5*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+42*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)*Ellip
ticE(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^2*c-21*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+5*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^2-42*A*c^3*x^4+10*a*B*c^2*x^3-
70*a*A*c^2*x^2+14*a^2*B*c*x-24*A*a^2*c)/a^3/e/(e*x)^(1/2)/c/(c*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(3/2)),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{{\left (c^{2} e x^{5} + 2 \, a c e x^{3} + a^{2} e x\right )} \sqrt{c x^{2} + a} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(3/2)),x, algorithm="fricas")

[Out]

integral((B*x + A)/((c^2*e*x^5 + 2*a*c*e*x^3 + a^2*e*x)*sqrt(c*x^2 + a)*sqrt(e*x
)), 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)/(e*x)**(3/2)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(3/2)),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(3/2)), x)