3.457 \(\int \frac{(A+B x) \left (a+c x^2\right )^{5/2}}{(e x)^{11/2}} \, dx\)

Optimal. Leaf size=375 \[ \frac{8 \sqrt [4]{a} c^{7/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 e^5 \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 c^2 \sqrt{a+c x^2} (7 A-5 B x)}{21 e^5 \sqrt{e x}}-\frac{4 c \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{63 e^3 (e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}+\frac{16 A c^{5/2} x \sqrt{a+c x^2}}{3 e^5 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{16 \sqrt [4]{a} A c^{9/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 e^5 \sqrt{e x} \sqrt{a+c x^2}} \]

[Out]

(-8*c^2*(7*A - 5*B*x)*Sqrt[a + c*x^2])/(21*e^5*Sqrt[e*x]) + (16*A*c^(5/2)*x*Sqrt
[a + c*x^2])/(3*e^5*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*c*(7*A + 15*B*x)*(a +
c*x^2)^(3/2))/(63*e^3*(e*x)^(5/2)) - (2*(7*A + 9*B*x)*(a + c*x^2)^(5/2))/(63*e*(
e*x)^(9/2)) - (16*a^(1/4)*A*c^(9/4)*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])
/(3*e^5*Sqrt[e*x]*Sqrt[a + c*x^2]) + (8*a^(1/4)*(5*Sqrt[a]*B + 7*A*Sqrt[c])*c^(7
/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elli
pticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(21*e^5*Sqrt[e*x]*Sqrt[a + c*x^
2])

_______________________________________________________________________________________

Rubi [A]  time = 1.01292, antiderivative size = 375, 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 \[ \frac{8 \sqrt [4]{a} c^{7/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 \sqrt{a} B+7 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 e^5 \sqrt{e x} \sqrt{a+c x^2}}-\frac{8 c^2 \sqrt{a+c x^2} (7 A-5 B x)}{21 e^5 \sqrt{e x}}-\frac{4 c \left (a+c x^2\right )^{3/2} (7 A+15 B x)}{63 e^3 (e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2} (7 A+9 B x)}{63 e (e x)^{9/2}}+\frac{16 A c^{5/2} x \sqrt{a+c x^2}}{3 e^5 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{16 \sqrt [4]{a} A c^{9/4} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 e^5 \sqrt{e x} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(5/2))/(e*x)^(11/2),x]

[Out]

(-8*c^2*(7*A - 5*B*x)*Sqrt[a + c*x^2])/(21*e^5*Sqrt[e*x]) + (16*A*c^(5/2)*x*Sqrt
[a + c*x^2])/(3*e^5*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (4*c*(7*A + 15*B*x)*(a +
c*x^2)^(3/2))/(63*e^3*(e*x)^(5/2)) - (2*(7*A + 9*B*x)*(a + c*x^2)^(5/2))/(63*e*(
e*x)^(9/2)) - (16*a^(1/4)*A*c^(9/4)*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])
/(3*e^5*Sqrt[e*x]*Sqrt[a + c*x^2]) + (8*a^(1/4)*(5*Sqrt[a]*B + 7*A*Sqrt[c])*c^(7
/4)*Sqrt[x]*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*Elli
pticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(21*e^5*Sqrt[e*x]*Sqrt[a + c*x^
2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 139.175, size = 366, normalized size = 0.98 \[ - \frac{16 A \sqrt [4]{a} c^{\frac{9}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 e^{5} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{16 A c^{\frac{5}{2}} x \sqrt{a + c x^{2}}}{3 e^{5} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{8 \sqrt [4]{a} c^{\frac{7}{4}} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (7 A \sqrt{c} + 5 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{21 e^{5} \sqrt{e x} \sqrt{a + c x^{2}}} - \frac{16 c^{2} \left (\frac{63 A}{2} - \frac{45 B x}{2}\right ) \sqrt{a + c x^{2}}}{189 e^{5} \sqrt{e x}} - \frac{8 c \left (\frac{21 A}{2} + \frac{45 B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{3}{2}}}{189 e^{3} \left (e x\right )^{\frac{5}{2}}} - \frac{4 \left (\frac{7 A}{2} + \frac{9 B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{63 e \left (e x\right )^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(5/2)/(e*x)**(11/2),x)

[Out]

-16*A*a**(1/4)*c**(9/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqr
t(a) + sqrt(c)*x)*elliptic_e(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(3*e**5*sqr
t(e*x)*sqrt(a + c*x**2)) + 16*A*c**(5/2)*x*sqrt(a + c*x**2)/(3*e**5*sqrt(e*x)*(s
qrt(a) + sqrt(c)*x)) + 8*a**(1/4)*c**(7/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a) +
sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(7*A*sqrt(c) + 5*B*sqrt(a))*elliptic_f(2*at
an(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(21*e**5*sqrt(e*x)*sqrt(a + c*x**2)) - 16*c*
*2*(63*A/2 - 45*B*x/2)*sqrt(a + c*x**2)/(189*e**5*sqrt(e*x)) - 8*c*(21*A/2 + 45*
B*x/2)*(a + c*x**2)**(3/2)/(189*e**3*(e*x)**(5/2)) - 4*(7*A/2 + 9*B*x/2)*(a + c*
x**2)**(5/2)/(63*e*(e*x)**(9/2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.35538, size = 259, normalized size = 0.69 \[ \frac{\sqrt{e x} \left (-2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (a^2 (7 A+9 B x)+4 a c x^2 (7 A+12 B x)-21 c^2 x^4 (3 A+B x)\right )+48 \sqrt{a} c^2 x^{11/2} \sqrt{\frac{a}{c x^2}+1} \left (7 A \sqrt{c}+5 i \sqrt{a} B\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-336 \sqrt{a} A c^{5/2} x^{11/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{63 e^6 x^5 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(5/2))/(e*x)^(11/2),x]

[Out]

(Sqrt[e*x]*(-2*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(-21*c^2*x^4*(3*A + B*x) +
a^2*(7*A + 9*B*x) + 4*a*c*x^2*(7*A + 12*B*x)) - 336*Sqrt[a]*A*c^(5/2)*Sqrt[1 + a
/(c*x^2)]*x^(11/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] +
48*Sqrt[a]*((5*I)*Sqrt[a]*B + 7*A*Sqrt[c])*c^2*Sqrt[1 + a/(c*x^2)]*x^(11/2)*Ell
ipticF[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(63*Sqrt[(I*Sqrt[a])/
Sqrt[c]]*e^6*x^5*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.052, size = 366, normalized size = 1. \[{\frac{2}{63\,{x}^{4}{e}^{5}} \left ( 168\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{4}a{c}^{2}-84\,A\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ){x}^{4}a{c}^{2}+60\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) \sqrt{-ac}{x}^{4}ac+21\,B{c}^{3}{x}^{7}-105\,A{c}^{3}{x}^{6}-27\,aB{c}^{2}{x}^{5}-133\,aA{c}^{2}{x}^{4}-57\,{a}^{2}Bc{x}^{3}-35\,{a}^{2}Ac{x}^{2}-9\,{a}^{3}Bx-7\,A{a}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(5/2)/(e*x)^(11/2),x)

[Out]

2/63/x^4*(168*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^2-84*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+60*
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+21*B*c^3*x^7-105*A*c^3*x^6-27*a*B*c^2*x
^5-133*a*A*c^2*x^4-57*a^2*B*c*x^3-35*a^2*A*c*x^2-9*a^3*B*x-7*A*a^3)/(c*x^2+a)^(1
/2)/e^5/(e*x)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(11/2), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B c^{2} x^{5} + A c^{2} x^{4} + 2 \, B a c x^{3} + 2 \, A a c x^{2} + B a^{2} x + A a^{2}\right )} \sqrt{c x^{2} + a}}{\sqrt{e x} e^{5} x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(11/2),x, algorithm="fricas")

[Out]

integral((B*c^2*x^5 + A*c^2*x^4 + 2*B*a*c*x^3 + 2*A*a*c*x^2 + B*a^2*x + A*a^2)*s
qrt(c*x^2 + a)/(sqrt(e*x)*e^5*x^5), 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((B*x+A)*(c*x**2+a)**(5/2)/(e*x)**(11/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (B x + A\right )}}{\left (e x\right )^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(11/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(5/2)*(B*x + A)/(e*x)^(11/2), x)