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

Optimal. Leaf size=378 \[ \frac{8 a^{7/4} \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (7 \sqrt{a} B+5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{16 a^{9/4} B \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 e^2 \sqrt{e x} \sqrt{a+c x^2}}+\frac{16 a^2 B \sqrt{c} x \sqrt{a+c x^2}}{3 e^2 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{8 a c \sqrt{e x} \sqrt{a+c x^2} (5 A+7 B x)}{21 e^3}-\frac{20 \left (a+c x^2\right )^{3/2} (7 a B-3 A c x)}{63 e^2 \sqrt{e x}}-\frac{2 \left (a+c x^2\right )^{5/2} (3 A-B x)}{9 e (e x)^{3/2}} \]

[Out]

(8*a*c*Sqrt[e*x]*(5*A + 7*B*x)*Sqrt[a + c*x^2])/(21*e^3) + (16*a^2*B*Sqrt[c]*x*S
qrt[a + c*x^2])/(3*e^2*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (20*(7*a*B - 3*A*c*x)*
(a + c*x^2)^(3/2))/(63*e^2*Sqrt[e*x]) - (2*(3*A - B*x)*(a + c*x^2)^(5/2))/(9*e*(
e*x)^(3/2)) - (16*a^(9/4)*B*c^(1/4)*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^2*Sqrt[e*x]*Sqrt[a + c*x^2]) + (8*a^(7/4)*(7*Sqrt[a]*B + 5*A*Sqrt[c])*c^(1
/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^2*Sqrt[e*x]*Sqrt[a + c*x^
2])

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Rubi [A]  time = 0.959149, antiderivative size = 378, 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 \[ \frac{8 a^{7/4} \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (7 \sqrt{a} B+5 A \sqrt{c}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 e^2 \sqrt{e x} \sqrt{a+c x^2}}-\frac{16 a^{9/4} B \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{3 e^2 \sqrt{e x} \sqrt{a+c x^2}}+\frac{16 a^2 B \sqrt{c} x \sqrt{a+c x^2}}{3 e^2 \sqrt{e x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{8 a c \sqrt{e x} \sqrt{a+c x^2} (5 A+7 B x)}{21 e^3}-\frac{20 \left (a+c x^2\right )^{3/2} (7 a B-3 A c x)}{63 e^2 \sqrt{e x}}-\frac{2 \left (a+c x^2\right )^{5/2} (3 A-B x)}{9 e (e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(8*a*c*Sqrt[e*x]*(5*A + 7*B*x)*Sqrt[a + c*x^2])/(21*e^3) + (16*a^2*B*Sqrt[c]*x*S
qrt[a + c*x^2])/(3*e^2*Sqrt[e*x]*(Sqrt[a] + Sqrt[c]*x)) - (20*(7*a*B - 3*A*c*x)*
(a + c*x^2)^(3/2))/(63*e^2*Sqrt[e*x]) - (2*(3*A - B*x)*(a + c*x^2)^(5/2))/(9*e*(
e*x)^(3/2)) - (16*a^(9/4)*B*c^(1/4)*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^2*Sqrt[e*x]*Sqrt[a + c*x^2]) + (8*a^(7/4)*(7*Sqrt[a]*B + 5*A*Sqrt[c])*c^(1
/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^2*Sqrt[e*x]*Sqrt[a + c*x^
2])

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Rubi in Sympy [A]  time = 127.088, size = 371, normalized size = 0.98 \[ - \frac{16 B a^{\frac{9}{4}} \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{3 e^{2} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{16 B a^{2} \sqrt{c} x \sqrt{a + c x^{2}}}{3 e^{2} \sqrt{e x} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{8 a^{\frac{7}{4}} \sqrt [4]{c} \sqrt{x} \sqrt{\frac{a + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (5 A \sqrt{c} + 7 B \sqrt{a}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{21 e^{2} \sqrt{e x} \sqrt{a + c x^{2}}} + \frac{16 a c \sqrt{e x} \left (\frac{45 A}{2} + \frac{63 B x}{2}\right ) \sqrt{a + c x^{2}}}{189 e^{3}} - \frac{4 \left (\frac{9 A}{2} - \frac{3 B x}{2}\right ) \left (a + c x^{2}\right )^{\frac{5}{2}}}{27 e \left (e x\right )^{\frac{3}{2}}} - \frac{40 \left (a + c x^{2}\right )^{\frac{3}{2}} \left (- \frac{9 A c x}{2} + \frac{21 B a}{2}\right )}{189 e^{2} \sqrt{e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*B*a**(9/4)*c**(1/4)*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**2*sqr
t(e*x)*sqrt(a + c*x**2)) + 16*B*a**2*sqrt(c)*x*sqrt(a + c*x**2)/(3*e**2*sqrt(e*x
)*(sqrt(a) + sqrt(c)*x)) + 8*a**(7/4)*c**(1/4)*sqrt(x)*sqrt((a + c*x**2)/(sqrt(a
) + sqrt(c)*x)**2)*(sqrt(a) + sqrt(c)*x)*(5*A*sqrt(c) + 7*B*sqrt(a))*elliptic_f(
2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(21*e**2*sqrt(e*x)*sqrt(a + c*x**2)) + 1
6*a*c*sqrt(e*x)*(45*A/2 + 63*B*x/2)*sqrt(a + c*x**2)/(189*e**3) - 4*(9*A/2 - 3*B
*x/2)*(a + c*x**2)**(5/2)/(27*e*(e*x)**(3/2)) - 40*(a + c*x**2)**(3/2)*(-9*A*c*x
/2 + 21*B*a/2)/(189*e**2*sqrt(e*x))

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Mathematica [C]  time = 0.8831, size = 253, normalized size = 0.67 \[ \frac{x \left (-336 a^{5/2} B \sqrt{c} x^{5/2} \sqrt{\frac{a}{c x^2}+1} E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )+2 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} \left (a+c x^2\right ) \left (-21 a^2 (A-5 B x)+4 a c x^2 (12 A+7 B x)+c^2 x^4 (9 A+7 B x)\right )+48 a^2 \sqrt{c} x^{5/2} \sqrt{\frac{a}{c x^2}+1} \left (7 \sqrt{a} B+5 i A \sqrt{c}\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{63 \sqrt{\frac{i \sqrt{a}}{\sqrt{c}}} (e x)^{5/2} \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(a + c*x^2)*(-21*a^2*(A - 5*B*x) + c^2*x^4*(9*A
+ 7*B*x) + 4*a*c*x^2*(12*A + 7*B*x)) - 336*a^(5/2)*B*Sqrt[c]*Sqrt[1 + a/(c*x^2)]
*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1] + 48*a^2*(7
*Sqrt[a]*B + (5*I)*A*Sqrt[c])*Sqrt[c]*Sqrt[1 + a/(c*x^2)]*x^(5/2)*EllipticF[I*Ar
cSinh[Sqrt[(I*Sqrt[a])/Sqrt[c]]/Sqrt[x]], -1]))/(63*Sqrt[(I*Sqrt[a])/Sqrt[c]]*(e
*x)^(5/2)*Sqrt[a + c*x^2])

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Maple [A]  time = 0.029, size = 359, normalized size = 1. \[{\frac{2}{63\,x{e}^{2}} \left ( 7\,B{c}^{3}{x}^{7}+60\,A\sqrt{-ac}\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) x{a}^{2}+9\,A{c}^{3}{x}^{6}-84\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) x{a}^{3}+168\,B\sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-ac}}{\sqrt{-ac}}}}\sqrt{-{\frac{cx}{\sqrt{-ac}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-ac}}{\sqrt{-ac}}}},1/2\,\sqrt{2} \right ) x{a}^{3}+35\,aB{c}^{2}{x}^{5}+57\,aA{c}^{2}{x}^{4}-35\,{a}^{2}Bc{x}^{3}+27\,{a}^{2}Ac{x}^{2}-63\,{a}^{3}Bx-21\,A{a}^{3} \right ){\frac{1}{\sqrt{c{x}^{2}+a}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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