3.1485 \(\int \frac{(A+B x) \sqrt{d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=319 \[ \frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} (a B e+A c d) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a B-A c x)}{a c \sqrt{a+c x^2}}-\frac{A \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

[Out]

-(((a*B - A*c*x)*Sqrt[d + e*x])/(a*c*Sqrt[a + c*x^2])) - (A*Sqrt[d + e*x]*Sqrt[1
+ (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)
/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c
]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + ((A*c*d + a*B*e)*Sqrt[(Sqrt[c]*(d + e*x))/
(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]
*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*c^(3/2)*
Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.683878, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} (a B e+A c d) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a B-A c x)}{a c \sqrt{a+c x^2}}-\frac{A \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

Antiderivative was successfully verified.

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

[Out]

-(((a*B - A*c*x)*Sqrt[d + e*x])/(a*c*Sqrt[a + c*x^2])) - (A*Sqrt[d + e*x]*Sqrt[1
+ (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)
/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c
]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + ((A*c*d + a*B*e)*Sqrt[(Sqrt[c]*(d + e*x))/
(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]
*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*c^(3/2)*
Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 124.009, size = 296, normalized size = 0.93 \[ - \frac{A \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{\sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (A c d + B a e\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{c^{\frac{3}{2}} \sqrt{- a} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{\sqrt{d + e x} \left (- A c x + B a\right )}{a c \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+a)**(3/2),x)

[Out]

-A*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a))
+ 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(sqrt(c)*sqrt(-a)*sqrt(sqrt(c)*sqrt(
-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)) + sqrt(sqrt(c)*sqrt
(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 + c*x**2/a)*(A*c*d + B*a*e)*e
lliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt
(-a)))/(c**(3/2)*sqrt(-a)*sqrt(a + c*x**2)*sqrt(d + e*x)) - sqrt(d + e*x)*(-A*c*
x + B*a)/(a*c*sqrt(a + c*x**2))

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Mathematica [C]  time = 7.67703, size = 542, normalized size = 1.7 \[ \frac{\sqrt{d+e x} (A c x-a B)}{a c \sqrt{a+c x^2}}-\frac{(d+e x)^{3/2} \left (-\frac{\sqrt{a} e \left (A \sqrt{c}+i \sqrt{a} B\right ) \sqrt{-\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} \sqrt{\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}+A \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (\frac{a e^2}{(d+e x)^2}+c \left (\frac{d}{d+e x}-1\right )^2\right )+\frac{A \sqrt{c} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{-\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} \sqrt{\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}\right )}{a c e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{a+\frac{c (d+e x)^2 \left (\frac{d}{d+e x}-1\right )^2}{e^2}}} \]

Antiderivative was successfully verified.

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

[Out]

((-(a*B) + A*c*x)*Sqrt[d + e*x])/(a*c*Sqrt[a + c*x^2]) - ((d + e*x)^(3/2)*(A*Sqr
t[-d - (I*Sqrt[a]*e)/Sqrt[c]]*((a*e^2)/(d + e*x)^2 + c*(-1 + d/(d + e*x))^2) + (
A*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[1 - d/(d + e*x) - (I*Sqrt[a]*e)/(Sqr
t[c]*(d + e*x))]*Sqrt[1 - d/(d + e*x) + (I*Sqrt[a]*e)/(Sqrt[c]*(d + e*x))]*Ellip
ticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*S
qrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x] - (Sqrt[a]*(I*Sqrt[a]*B + A*
Sqrt[c])*e*Sqrt[1 - d/(d + e*x) - (I*Sqrt[a]*e)/(Sqrt[c]*(d + e*x))]*Sqrt[1 - d/
(d + e*x) + (I*Sqrt[a]*e)/(Sqrt[c]*(d + e*x))]*EllipticF[I*ArcSinh[Sqrt[-d - (I*
Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqr
t[a]*e)])/Sqrt[d + e*x]))/(a*c*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[a + (c*(d
+ e*x)^2*(-1 + d/(d + e*x))^2)/e^2])

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Maple [B]  time = 0.062, size = 979, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(1/2)/(c*x^2+a)^(3/2),x)

[Out]

(-A*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-
a*c)^(1/2)*e+c*d))^(1/2))*a*c*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x
+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)
*e-c*d))^(1/2)-A*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/
2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c*d*e*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1
/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c
)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+A*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c
*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c*e^2*(-(e*x+d)
*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2
)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+A*EllipticE((-(e*x+d)*c/((-a
*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^2*
d^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)
*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+B*EllipticF((-(
e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d)
)^(1/2))*a*c*d*e*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/
((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-B*
EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)
^(1/2)*e+c*d))^(1/2))*a*e^2*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)
*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c
)^(1/2)*e-c*d))^(1/2)+A*x^2*c^2*e^2+A*c^2*d*e*x-B*x*a*c*e^2-B*a*c*d*e)*(e*x+d)^(
1/2)*(c*x^2+a)^(1/2)/e/(c*e*x^3+c*d*x^2+a*e*x+a*d)/a/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+a)**(3/2),x)

[Out]

Integral((A + B*x)*sqrt(d + e*x)/(a + c*x**2)**(3/2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError