3.678 \(\int \frac{1}{\sqrt{d+e x} \left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=331 \[ \frac{\sqrt{d+e x} (a e+c d x)}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{\sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}} \]

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) - (Sqrt[c]*d*S
qrt[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]*(c*d^2 + a*e^2)*Sqrt[
(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (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]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.76636, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{\sqrt{d+e x} (a e+c d x)}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{\sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[d + e*x]*(a + c*x^2)^(3/2)),x]

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(a*(c*d^2 + a*e^2)*Sqrt[a + c*x^2]) - (Sqrt[c]*d*S
qrt[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]*(c*d^2 + a*e^2)*Sqrt[
(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (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]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 124.194, size = 304, normalized size = 0.92 \[ - \frac{\sqrt{c} d \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} \sqrt{- a} \sqrt{a + c x^{2}} \sqrt{d + e x}} + \frac{\sqrt{d + e x} \left (a e + c d x\right )}{a \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c)*d*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*s
qrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(sqrt(-a)*sqrt(sqrt(c)*sqrt(
-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)*(a*e**2 + c*d**2)) +
sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 + c*x**2/a)
*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sq
rt(-a)))/(sqrt(c)*sqrt(-a)*sqrt(a + c*x**2)*sqrt(d + e*x)) + sqrt(d + e*x)*(a*e
+ c*d*x)/(a*sqrt(a + c*x**2)*(a*e**2 + c*d**2))

_______________________________________________________________________________________

Mathematica [C]  time = 1.74324, size = 430, normalized size = 1.3 \[ \frac{e x \left (\sqrt{c} d-i \sqrt{a} e\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}+\sqrt{a} e (d+e x)^{3/2} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+i \sqrt{c} d (d+e x)^{3/2} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{a e \sqrt{a+c x^2} \sqrt{d+e x} \left (\sqrt{c} d-i \sqrt{a} e\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[d + e*x]*(a + c*x^2)^(3/2)),x]

[Out]

(e*(Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*x + I*Sqrt[c]*d*Sq
rt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)
/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[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*Sqrt[a]*e)] + Sqrt[a]
*e*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] -
e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sq
rt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/(a*
e*(Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[d + e*x]*Sqrt[
a + c*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.067, size = 696, normalized size = 2.1 \[{\frac{1}{ \left ( a{e}^{2}+c{d}^{2} \right ) ace \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ) } \left ( -\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}{\it EllipticF} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) \sqrt{-ac}a{e}^{3}-\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}{\it EllipticF} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) \sqrt{-ac}c{d}^{2}e+\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}{\it EllipticE} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) acd{e}^{2}+\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}{\it EllipticE} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ){c}^{2}{d}^{3}+{x}^{2}{c}^{2}d{e}^{2}+xac{e}^{3}+x{c}^{2}{d}^{2}e+ad{e}^{2}c \right ) \sqrt{ex+d}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-(-(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)*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)^(1/2)*a*e^3-(-(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)*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)^(1/2)*c*d^2*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)*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*d*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)*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^3+x^2*c^2*
d*e^2+x*a*c*e^3+x*c^2*d^2*e+a*d*e^2*c)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e/c/a/(a*e^
2+c*d^2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(3/2)*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^(3/2)*sqrt(e*x + d)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c x^{2} + a\right )}^{\frac{3}{2}} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(3/2)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

integral(1/((c*x^2 + a)^(3/2)*sqrt(e*x + d)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right )^{\frac{3}{2}} \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + c*x**2)**(3/2)*sqrt(d + e*x)), x)

_______________________________________________________________________________________

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(1/((c*x^2 + a)^(3/2)*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError