3.484 \(\int \frac{A+B x}{(e x)^{7/2} \left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=432 \[ -\frac{c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+77 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{77 A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{77 A c^{3/2} x \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}+\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}} \]

[Out]

(A + B*x)/(3*a*e*(e*x)^(5/2)*(a + c*x^2)^(3/2)) + (11*A + 9*B*x)/(6*a^2*e*(e*x)^
(5/2)*Sqrt[a + c*x^2]) - (77*A*Sqrt[a + c*x^2])/(30*a^3*e*(e*x)^(5/2)) - (5*B*Sq
rt[a + c*x^2])/(2*a^3*e^2*(e*x)^(3/2)) + (77*A*c*Sqrt[a + c*x^2])/(10*a^4*e^3*Sq
rt[e*x]) - (77*A*c^(3/2)*x*Sqrt[a + c*x^2])/(10*a^4*e^3*Sqrt[e*x]*(Sqrt[a] + Sqr
t[c]*x)) + (77*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])/(10*a^(15/
4)*e^3*Sqrt[e*x]*Sqrt[a + c*x^2]) - ((25*Sqrt[a]*B + 77*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*A
rcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(20*a^(15/4)*e^3*Sqrt[e*x]*Sqrt[a + c*x^
2])

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Rubi [A]  time = 1.38522, antiderivative size = 432, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{c^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (25 \sqrt{a} B+77 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}+\frac{77 A c^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{15/4} e^3 \sqrt{e x} \sqrt{a+c x^2}}-\frac{77 A c^{3/2} x \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{77 A c \sqrt{a+c x^2}}{10 a^4 e^3 \sqrt{e x}}-\frac{77 A \sqrt{a+c x^2}}{30 a^3 e (e x)^{5/2}}-\frac{5 B \sqrt{a+c x^2}}{2 a^3 e^2 (e x)^{3/2}}+\frac{11 A+9 B x}{6 a^2 e (e x)^{5/2} \sqrt{a+c x^2}}+\frac{A+B x}{3 a e (e x)^{5/2} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((e*x)^(7/2)*(a + c*x^2)^(5/2)),x]

[Out]

(A + B*x)/(3*a*e*(e*x)^(5/2)*(a + c*x^2)^(3/2)) + (11*A + 9*B*x)/(6*a^2*e*(e*x)^
(5/2)*Sqrt[a + c*x^2]) - (77*A*Sqrt[a + c*x^2])/(30*a^3*e*(e*x)^(5/2)) - (5*B*Sq
rt[a + c*x^2])/(2*a^3*e^2*(e*x)^(3/2)) + (77*A*c*Sqrt[a + c*x^2])/(10*a^4*e^3*Sq
rt[e*x]) - (77*A*c^(3/2)*x*Sqrt[a + c*x^2])/(10*a^4*e^3*Sqrt[e*x]*(Sqrt[a] + Sqr
t[c]*x)) + (77*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])/(10*a^(15/
4)*e^3*Sqrt[e*x]*Sqrt[a + c*x^2]) - ((25*Sqrt[a]*B + 77*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*A
rcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(20*a^(15/4)*e^3*Sqrt[e*x]*Sqrt[a + c*x^
2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x)**(7/2)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 1.15284, size = 260, normalized size = 0.6 \[ \frac{x \left (-\sqrt{a} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (4 a^2 (3 A+5 B x)+3 a c x^2 (33 A+35 B x)+c^2 x^4 (77 A+75 B x)\right )-3 c x^{7/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) \left (77 A \sqrt{c}+25 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+231 A c^{3/2} x^{7/2} \sqrt{\frac{a}{c x^2}+1} \left (a+c x^2\right ) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{30 a^{7/2} \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{7/2} \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((e*x)^(7/2)*(a + c*x^2)^(5/2)),x]

[Out]

(x*(-(Sqrt[a]*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(4*a^2*(3*A + 5*B*x) + 3*a*c*x^2*(33*A +
35*B*x) + c^2*x^4*(77*A + 75*B*x))) + 231*A*c^(3/2)*Sqrt[1 + a/(c*x^2)]*x^(7/2)
*(a + c*x^2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] - 3*((2
5*I)*Sqrt[a]*B + 77*A*Sqrt[c])*c*Sqrt[1 + a/(c*x^2)]*x^(7/2)*(a + c*x^2)*Ellipti
cF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(30*a^(7/2)*Sqrt[(I*Sqrt[
a])/Sqrt[c]]*(e*x)^(7/2)*(a + c*x^2)^(3/2))

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Maple [A]  time = 0.035, size = 632, normalized size = 1.5 \[ -{\frac{1}{60\,{x}^{2}{a}^{4}{e}^{3}} \left ( 462\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{4}a{c}^{2}-231\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{4}a{c}^{2}+75\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{4}ac+462\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c-231\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}c+75\,B\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{2}{a}^{2}-462\,A{c}^{3}{x}^{6}+150\,aB{c}^{2}{x}^{5}-770\,aA{c}^{2}{x}^{4}+210\,{a}^{2}Bc{x}^{3}-264\,{a}^{2}Ac{x}^{2}+40\,{a}^{3}Bx+24\,A{a}^{3} \right ){\frac{1}{\sqrt{ex}}} \left ( c{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x)^(7/2)/(c*x^2+a)^(5/2),x)

[Out]

-1/60*(462*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^4*a*c^2-231*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^4*a*c^2+75*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^4*a*c+462*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^2*c-231*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^2*c+75*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^2-462*A*c^3
*x^6+150*a*B*c^2*x^5-770*a*A*c^2*x^4+210*a^2*B*c*x^3-264*a^2*A*c*x^2+40*a^3*B*x+
24*A*a^3)/x^2/a^4/e^3/(e*x)^(1/2)/(c*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(7/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{{\left (c^{2} e^{3} x^{7} + 2 \, a c e^{3} x^{5} + a^{2} e^{3} x^{3}\right )} \sqrt{c x^{2} + a} \sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(7/2)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{{\left (c x^{2} + a\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + a)^(5/2)*(e*x)^(7/2)),x, algorithm="giac")

[Out]

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