3.458 \(\int \frac{(e x)^{7/2} (A+B x)}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=388 \[ \frac{a^{7/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (49 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{14 a^{9/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{14 a^2 B e^4 x \sqrt{a+c x^2}}{15 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{10 a A e^3 \sqrt{e x} \sqrt{a+c x^2}}{21 c^2}+\frac{2 A e (e x)^{5/2} \sqrt{a+c x^2}}{7 c}-\frac{14 a B e^2 (e x)^{3/2} \sqrt{a+c x^2}}{45 c^2}+\frac{2 B (e x)^{7/2} \sqrt{a+c x^2}}{9 c} \]

[Out]

(-10*a*A*e^3*Sqrt[e*x]*Sqrt[a + c*x^2])/(21*c^2) - (14*a*B*e^2*(e*x)^(3/2)*Sqrt[
a + c*x^2])/(45*c^2) + (2*A*e*(e*x)^(5/2)*Sqrt[a + c*x^2])/(7*c) + (2*B*(e*x)^(7
/2)*Sqrt[a + c*x^2])/(9*c) + (14*a^2*B*e^4*x*Sqrt[a + c*x^2])/(15*c^(5/2)*Sqrt[e
*x]*(Sqrt[a] + Sqrt[c]*x)) - (14*a^(9/4)*B*e^4*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqr
t[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1
/4)], 1/2])/(15*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (a^(7/4)*(49*Sqrt[a]*B + 2
5*A*Sqrt[c])*e^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])/(105*c^(11/4)*Sqrt
[e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.14557, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{a^{7/4} e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (49 \sqrt{a} B+25 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{105 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}-\frac{14 a^{9/4} B e^4 \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{11/4} \sqrt{e x} \sqrt{a+c x^2}}+\frac{14 a^2 B e^4 x \sqrt{a+c x^2}}{15 c^{5/2} \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{10 a A e^3 \sqrt{e x} \sqrt{a+c x^2}}{21 c^2}+\frac{2 A e (e x)^{5/2} \sqrt{a+c x^2}}{7 c}-\frac{14 a B e^2 (e x)^{3/2} \sqrt{a+c x^2}}{45 c^2}+\frac{2 B (e x)^{7/2} \sqrt{a+c x^2}}{9 c} \]

Antiderivative was successfully verified.

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

[Out]

(-10*a*A*e^3*Sqrt[e*x]*Sqrt[a + c*x^2])/(21*c^2) - (14*a*B*e^2*(e*x)^(3/2)*Sqrt[
a + c*x^2])/(45*c^2) + (2*A*e*(e*x)^(5/2)*Sqrt[a + c*x^2])/(7*c) + (2*B*(e*x)^(7
/2)*Sqrt[a + c*x^2])/(9*c) + (14*a^2*B*e^4*x*Sqrt[a + c*x^2])/(15*c^(5/2)*Sqrt[e
*x]*(Sqrt[a] + Sqrt[c]*x)) - (14*a^(9/4)*B*e^4*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqr
t[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1
/4)], 1/2])/(15*c^(11/4)*Sqrt[e*x]*Sqrt[a + c*x^2]) + (a^(7/4)*(49*Sqrt[a]*B + 2
5*A*Sqrt[c])*e^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])/(105*c^(11/4)*Sqrt
[e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 153.685, size = 367, normalized size = 0.95 \[ - \frac{10 A a e^{3} \sqrt{e x} \sqrt{a + c x^{2}}}{21 c^{2}} + \frac{2 A e \left (e x\right )^{\frac{5}{2}} \sqrt{a + c x^{2}}}{7 c} - \frac{14 B a^{\frac{9}{4}} e^{4} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{11}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{14 B a^{2} e^{4} x \sqrt{a + c x^{2}}}{15 c^{\frac{5}{2}} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} - \frac{14 B a e^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{a + c x^{2}}}{45 c^{2}} + \frac{2 B \left (e x\right )^{\frac{7}{2}} \sqrt{a + c x^{2}}}{9 c} + \frac{a^{\frac{7}{4}} e^{4} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (25 A \sqrt{c} + 49 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{105 c^{\frac{11}{4}} \sqrt{e x} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(B*x+A)/(c*x**2+a)**(1/2),x)

[Out]

-10*A*a*e**3*sqrt(e*x)*sqrt(a + c*x**2)/(21*c**2) + 2*A*e*(e*x)**(5/2)*sqrt(a +
c*x**2)/(7*c) - 14*B*a**(9/4)*e**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)/
(15*c**(11/4)*sqrt(e*x)*sqrt(a + c*x**2)) + 14*B*a**2*e**4*x*sqrt(a + c*x**2)/(1
5*c**(5/2)*sqrt(e*x)*(sqrt(a) + sqrt(c)*x)) - 14*B*a*e**2*(e*x)**(3/2)*sqrt(a +
c*x**2)/(45*c**2) + 2*B*(e*x)**(7/2)*sqrt(a + c*x**2)/(9*c) + a**(7/4)*e**4*sqrt
(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(25*A*sqrt
(c) + 49*B*sqrt(a))*elliptic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(105*c**(
11/4)*sqrt(e*x)*sqrt(a + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.922831, size = 251, normalized size = 0.65 \[ \frac{2 e^4 \left (-147 a^{5/2} B \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (147 a^2 B-a c x (75 A+49 B x)+5 c^2 x^3 (9 A+7 B x)\right )+3 a^2 \sqrt{c} x^{3/2} \sqrt{\frac{a}{c x^2}+1} \left (49 \sqrt{a} B+25 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{315 c^3 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e^4*(Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(147*a^2*B + 5*c^2*x^3*(9*A + 7*B*
x) - a*c*x*(75*A + 49*B*x)) - 147*a^(5/2)*B*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*
EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + 3*a^2*(49*Sqrt[a]*
B + (25*I)*A*Sqrt[c])*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sq
rt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(315*Sqrt[(I*Sqrt[a])/Sqrt[c]]*c^3*Sqrt[
e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.04, size = 344, normalized size = 0.9 \[{\frac{{e}^{3}}{315\,x{c}^{3}}\sqrt{ex} \left ( 70\,B{c}^{3}{x}^{6}+75\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{a}^{2}+90\,A{c}^{3}{x}^{5}-147\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}+294\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){a}^{3}-28\,aB{c}^{2}{x}^{4}-60\,aA{c}^{2}{x}^{3}-98\,{a}^{2}Bc{x}^{2}-150\,{a}^{2}Acx \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(B*x+A)/(c*x^2+a)^(1/2),x)

[Out]

1/315*e^3/x*(e*x)^(1/2)/(c*x^2+a)^(1/2)/c^3*(70*B*c^3*x^6+75*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)*a^2+90*A*c^3*x^5-147*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)*Elliptic
F(((c*x+(-a*c)^(1/2))/(-a*c)^(1/2))^(1/2),1/2*2^(1/2))*a^3+294*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
))*a^3-28*a*B*c^2*x^4-60*a*A*c^2*x^3-98*a^2*B*c*x^2-150*a^2*A*c*x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{3} x^{4} + A e^{3} x^{3}\right )} \sqrt{e x}}{\sqrt{c x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*e^3*x^4 + A*e^3*x^3)*sqrt(e*x)/sqrt(c*x^2 + a), x)

_______________________________________________________________________________________

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)**(7/2)*(B*x+A)/(c*x**2+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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