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

Optimal. Leaf size=393 \[ -\frac{c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+63 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 a^{11/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{21 A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{11/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{21 A c^{3/2} x \sqrt{a+c x^2}}{5 a^3 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{21 A c \sqrt{a+c x^2}}{5 a^3 e^3 \sqrt{e x}}-\frac{7 A \sqrt{a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac{A+B x}{a e (e x)^{5/2} \sqrt{a+c x^2}} \]

[Out]

(A + B*x)/(a*e*(e*x)^(5/2)*Sqrt[a + c*x^2]) - (7*A*Sqrt[a + c*x^2])/(5*a^2*e*(e*
x)^(5/2)) - (5*B*Sqrt[a + c*x^2])/(3*a^2*e^2*(e*x)^(3/2)) + (21*A*c*Sqrt[a + c*x
^2])/(5*a^3*e^3*Sqrt[e*x]) - (21*A*c^(3/2)*x*Sqrt[a + c*x^2])/(5*a^3*e^3*Sqrt[e*
x]*(Sqrt[a] + Sqrt[c]*x)) + (21*A*c^(5/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a
+ c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)],
1/2])/(5*a^(11/4)*e^3*Sqrt[e*x]*Sqrt[a + c*x^2]) - ((25*Sqrt[a]*B + 63*A*Sqrt[c
])*c^(3/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^
2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(30*a^(11/4)*e^3*Sqrt[e*
x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.18852, antiderivative size = 393, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+63 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 a^{11/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{21 A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{11/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{21 A c^{3/2} x \sqrt{a+c x^2}}{5 a^3 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{21 A c \sqrt{a+c x^2}}{5 a^3 e^3 \sqrt{e x}}-\frac{7 A \sqrt{a+c x^2}}{5 a^2 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{3 a^2 e^2 (e x)^{3/2}}+\frac{A+B x}{a e (e x)^{5/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(A + B*x)/(a*e*(e*x)^(5/2)*Sqrt[a + c*x^2]) - (7*A*Sqrt[a + c*x^2])/(5*a^2*e*(e*
x)^(5/2)) - (5*B*Sqrt[a + c*x^2])/(3*a^2*e^2*(e*x)^(3/2)) + (21*A*c*Sqrt[a + c*x
^2])/(5*a^3*e^3*Sqrt[e*x]) - (21*A*c^(3/2)*x*Sqrt[a + c*x^2])/(5*a^3*e^3*Sqrt[e*
x]*(Sqrt[a] + Sqrt[c]*x)) + (21*A*c^(5/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a
+ c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)],
1/2])/(5*a^(11/4)*e^3*Sqrt[e*x]*Sqrt[a + c*x^2]) - ((25*Sqrt[a]*B + 63*A*Sqrt[c
])*c^(3/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^
2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(30*a^(11/4)*e^3*Sqrt[e*
x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 174.173, size = 369, normalized size = 0.94 \[ - \frac{7 A \sqrt{a + c x^{2}}}{5 a^{2} e \left (e x\right )^{\frac{5}{2}}} - \frac{21 A c^{\frac{3}{2}} x \sqrt{a + c x^{2}}}{5 a^{3} e^{3} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{21 A c \sqrt{a + c x^{2}}}{5 a^{3} e^{3} \sqrt{e x}} + \frac{21 A c^{\frac{5}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{11}{4}} e^{3} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{5 B \sqrt{a + c x^{2}}}{3 a^{2} e^{2} \left (e x\right )^{\frac{3}{2}}} + \frac{A + B x}{a e \left (e x\right )^{\frac{5}{2}} \sqrt{a + c x^{2}}} - \frac{c^{\frac{3}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (63 A \sqrt{c} + 25 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{30 a^{\frac{11}{4}} e^{3} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-7*A*sqrt(a + c*x**2)/(5*a**2*e*(e*x)**(5/2)) - 21*A*c**(3/2)*x*sqrt(a + c*x**2)
/(5*a**3*e**3*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) + 21*A*c*sqrt(a + c*x**2)/(5*a**3
*e**3*sqrt(e*x)) + 21*A*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**(11/4)*e**3*sqrt(e*x)*sqrt(a + c*x**2)) - 5*B*sqrt(a + c*x**2)/(3*a**2*e**2*
(e*x)**(3/2)) + (A + B*x)/(a*e*(e*x)**(5/2)*sqrt(a + c*x**2)) - c**(3/4)*sqrt(x)
*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(63*A*sqrt(c)
+ 25*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(30*a**(11/4
)*e**3*sqrt(e*x)*sqrt(a + c*x**2))

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Mathematica [C]  time = 0.771731, size = 226, normalized size = 0.58 \[ \frac{x \left (-c x^{7/2} \sqrt{\frac{a}{c x^2}+1} \left (63 A \sqrt{c}+25 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-\sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (6 a A+10 a B x+21 A c x^2+25 B c x^3\right )+63 A c^{3/2} x^{7/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{15 a^{5/2} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{7/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(-(Sqrt[a]*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(6*a*A + 10*a*B*x + 21*A*c*x^2 + 25*B*c*
x^3)) + 63*A*c^(3/2)*Sqrt[1 + a/(c*x^2)]*x^(7/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqr
t[a])/Sqrt[c]]/Sqrt[x]], -1] - ((25*I)*Sqrt[a]*B + 63*A*Sqrt[c])*c*Sqrt[1 + a/(c
*x^2)]*x^(7/2)*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(15
*a^(5/2)*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(e*x)^(7/2)*Sqrt[a + c*x^2])

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Maple [A]  time = 0.031, size = 331, normalized size = 0.8 \[ -{\frac{1}{30\,{x}^{2}{e}^{3}{a}^{3}} \left ( 126\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac-63\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}ac+25\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{2}a-126\,A{c}^{2}{x}^{4}+50\,aBc{x}^{3}-84\,aAc{x}^{2}+20\,{a}^{2}Bx+12\,A{a}^{2} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30/x^2*(126*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-63*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+25*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))*(-a*c)^(1/2)*x^2*a-126*A*c^2*x^4+50*a*B*c*x^3-84*a*A*c*x^2+20*a
^2*B*x+12*A*a^2)/(c*x^2+a)^(1/2)/e^3/(e*x)^(1/2)/a^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)/((c*e^3*x^5 + a*e^3*x^3)*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)**(7/2)/(c*x**2+a)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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