3.440 \(\int \frac{(c+d x)^{3/2}}{\sqrt{a+\frac{b}{x^2}}} \, dx\)

Optimal. Leaf size=406 \[ -\frac{2 \sqrt{b} c \sqrt{\frac{a x^2}{b}+1} \left (a c^2+b d^2\right ) \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{c+d x}}+\frac{2 \sqrt{b} \sqrt{\frac{a x^2}{b}+1} \sqrt{c+d x} \left (a c^2-3 b d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}}}+\frac{2 \left (a x^2+b\right ) (c+d x)^{3/2}}{5 a x \sqrt{a+\frac{b}{x^2}}}+\frac{2 c \left (a x^2+b\right ) \sqrt{c+d x}}{5 a x \sqrt{a+\frac{b}{x^2}}} \]

[Out]

(2*c*Sqrt[c + d*x]*(b + a*x^2))/(5*a*Sqrt[a + b/x^2]*x) + (2*(c + d*x)^(3/2)*(b
+ a*x^2))/(5*a*Sqrt[a + b/x^2]*x) + (2*Sqrt[b]*(a*c^2 - 3*b*d^2)*Sqrt[c + d*x]*S
qrt[1 + (a*x^2)/b]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[-a]*x)/Sqrt[b]]/Sqrt[2]], (-2
*Sqrt[-a]*Sqrt[b]*d)/(a*c - Sqrt[-a]*Sqrt[b]*d)])/(5*(-a)^(3/2)*d*Sqrt[a + b/x^2
]*x*Sqrt[(a*(c + d*x))/(a*c - Sqrt[-a]*Sqrt[b]*d)]) - (2*Sqrt[b]*c*(a*c^2 + b*d^
2)*Sqrt[(a*(c + d*x))/(a*c - Sqrt[-a]*Sqrt[b]*d)]*Sqrt[1 + (a*x^2)/b]*EllipticF[
ArcSin[Sqrt[1 - (Sqrt[-a]*x)/Sqrt[b]]/Sqrt[2]], (-2*Sqrt[-a]*Sqrt[b]*d)/(a*c - S
qrt[-a]*Sqrt[b]*d)])/(5*(-a)^(3/2)*d*Sqrt[a + b/x^2]*x*Sqrt[c + d*x])

_______________________________________________________________________________________

Rubi [A]  time = 1.13368, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 \sqrt{b} c \sqrt{\frac{a x^2}{b}+1} \left (a c^2+b d^2\right ) \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{c+d x}}+\frac{2 \sqrt{b} \sqrt{\frac{a x^2}{b}+1} \sqrt{c+d x} \left (a c^2-3 b d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{-a} x}{\sqrt{b}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-a} \sqrt{b} d}{a c-\sqrt{-a} \sqrt{b} d}\right )}{5 (-a)^{3/2} d x \sqrt{a+\frac{b}{x^2}} \sqrt{\frac{a (c+d x)}{a c-\sqrt{-a} \sqrt{b} d}}}+\frac{2 \left (a x^2+b\right ) (c+d x)^{3/2}}{5 a x \sqrt{a+\frac{b}{x^2}}}+\frac{2 c \left (a x^2+b\right ) \sqrt{c+d x}}{5 a x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*c*Sqrt[c + d*x]*(b + a*x^2))/(5*a*Sqrt[a + b/x^2]*x) + (2*(c + d*x)^(3/2)*(b
+ a*x^2))/(5*a*Sqrt[a + b/x^2]*x) + (2*Sqrt[b]*(a*c^2 - 3*b*d^2)*Sqrt[c + d*x]*S
qrt[1 + (a*x^2)/b]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[-a]*x)/Sqrt[b]]/Sqrt[2]], (-2
*Sqrt[-a]*Sqrt[b]*d)/(a*c - Sqrt[-a]*Sqrt[b]*d)])/(5*(-a)^(3/2)*d*Sqrt[a + b/x^2
]*x*Sqrt[(a*(c + d*x))/(a*c - Sqrt[-a]*Sqrt[b]*d)]) - (2*Sqrt[b]*c*(a*c^2 + b*d^
2)*Sqrt[(a*(c + d*x))/(a*c - Sqrt[-a]*Sqrt[b]*d)]*Sqrt[1 + (a*x^2)/b]*EllipticF[
ArcSin[Sqrt[1 - (Sqrt[-a]*x)/Sqrt[b]]/Sqrt[2]], (-2*Sqrt[-a]*Sqrt[b]*d)/(a*c - S
qrt[-a]*Sqrt[b]*d)])/(5*(-a)^(3/2)*d*Sqrt[a + b/x^2]*x*Sqrt[c + d*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 75.39, size = 357, normalized size = 0.88 \[ - \frac{2 \sqrt{b} c x \sqrt{\frac{a \left (c + d x\right )}{a c - \sqrt{b} d \sqrt{- a}}} \sqrt{a + \frac{b}{x^{2}}} \left (a c^{2} + b d^{2}\right ) \sqrt{\frac{a x^{2}}{b} + 1} F\left (\operatorname{asin}{\left (\sqrt{\frac{1}{2} - \frac{x \sqrt{- a}}{2 \sqrt{b}}} \right )}\middle | - \frac{2 \sqrt{b} d \sqrt{- a}}{a c - \sqrt{b} d \sqrt{- a}}\right )}{5 d \left (- a\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a x^{2} + b\right )} + \frac{2 \sqrt{b} x \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + d x} \left (a c^{2} - 3 b d^{2}\right ) \sqrt{\frac{a x^{2}}{b} + 1} E\left (\operatorname{asin}{\left (\sqrt{\frac{1}{2} - \frac{x \sqrt{- a}}{2 \sqrt{b}}} \right )}\middle | - \frac{2 \sqrt{b} d \sqrt{- a}}{a c - \sqrt{b} d \sqrt{- a}}\right )}{5 d \left (- a\right )^{\frac{3}{2}} \sqrt{\frac{a \left (c + d x\right )}{a c - \sqrt{b} d \sqrt{- a}}} \left (a x^{2} + b\right )} + \frac{2 c x \sqrt{a + \frac{b}{x^{2}}} \sqrt{c + d x}}{5 a} + \frac{2 x \sqrt{a + \frac{b}{x^{2}}} \left (c + d x\right )^{\frac{3}{2}}}{5 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b)*c*x*sqrt(a*(c + d*x)/(a*c - sqrt(b)*d*sqrt(-a)))*sqrt(a + b/x**2)*(a*
c**2 + b*d**2)*sqrt(a*x**2/b + 1)*elliptic_f(asin(sqrt(1/2 - x*sqrt(-a)/(2*sqrt(
b)))), -2*sqrt(b)*d*sqrt(-a)/(a*c - sqrt(b)*d*sqrt(-a)))/(5*d*(-a)**(3/2)*sqrt(c
+ d*x)*(a*x**2 + b)) + 2*sqrt(b)*x*sqrt(a + b/x**2)*sqrt(c + d*x)*(a*c**2 - 3*b
*d**2)*sqrt(a*x**2/b + 1)*elliptic_e(asin(sqrt(1/2 - x*sqrt(-a)/(2*sqrt(b)))), -
2*sqrt(b)*d*sqrt(-a)/(a*c - sqrt(b)*d*sqrt(-a)))/(5*d*(-a)**(3/2)*sqrt(a*(c + d*
x)/(a*c - sqrt(b)*d*sqrt(-a)))*(a*x**2 + b)) + 2*c*x*sqrt(a + b/x**2)*sqrt(c + d
*x)/(5*a) + 2*x*sqrt(a + b/x**2)*(c + d*x)**(3/2)/(5*a)

_______________________________________________________________________________________

Mathematica [C]  time = 4.59329, size = 540, normalized size = 1.33 \[ \frac{\sqrt{c+d x} \left (\frac{2 \left (a x^2+b\right ) (2 c+d x)}{a}+\frac{2 \left (\sqrt{a} (c+d x)^{3/2} \left (-i a^{3/2} c^3+a \sqrt{b} c^2 d+3 i \sqrt{a} b c d^2-3 b^{3/2} d^3\right ) \sqrt{\frac{d \left (x+\frac{i \sqrt{b}}{\sqrt{a}}\right )}{c+d x}} \sqrt{-\frac{-d x+\frac{i \sqrt{b} d}{\sqrt{a}}}{c+d x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}{\sqrt{c+d x}}\right )|\frac{\sqrt{a} c-i \sqrt{b} d}{\sqrt{a} c+i \sqrt{b} d}\right )+d^2 \sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}} \left (a^2 c^2 x^2+a b \left (c^2-3 d^2 x^2\right )-3 b^2 d^2\right )-\sqrt{a} \sqrt{b} d (c+d x)^{3/2} \left (4 i \sqrt{a} \sqrt{b} c d+a c^2-3 b d^2\right ) \sqrt{\frac{d \left (x+\frac{i \sqrt{b}}{\sqrt{a}}\right )}{c+d x}} \sqrt{-\frac{-d x+\frac{i \sqrt{b} d}{\sqrt{a}}}{c+d x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}{\sqrt{c+d x}}\right )|\frac{\sqrt{a} c-i \sqrt{b} d}{\sqrt{a} c+i \sqrt{b} d}\right )\right )}{a^2 d^2 (c+d x) \sqrt{-c-\frac{i \sqrt{b} d}{\sqrt{a}}}}\right )}{5 x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*((2*(2*c + d*x)*(b + a*x^2))/a + (2*(d^2*Sqrt[-c - (I*Sqrt[b]*d)/
Sqrt[a]]*(-3*b^2*d^2 + a^2*c^2*x^2 + a*b*(c^2 - 3*d^2*x^2)) + Sqrt[a]*((-I)*a^(3
/2)*c^3 + a*Sqrt[b]*c^2*d + (3*I)*Sqrt[a]*b*c*d^2 - 3*b^(3/2)*d^3)*Sqrt[(d*((I*S
qrt[b])/Sqrt[a] + x))/(c + d*x)]*Sqrt[-(((I*Sqrt[b]*d)/Sqrt[a] - d*x)/(c + d*x))
]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]/Sqrt[c +
d*x]], (Sqrt[a]*c - I*Sqrt[b]*d)/(Sqrt[a]*c + I*Sqrt[b]*d)] - Sqrt[a]*Sqrt[b]*d*
(a*c^2 + (4*I)*Sqrt[a]*Sqrt[b]*c*d - 3*b*d^2)*Sqrt[(d*((I*Sqrt[b])/Sqrt[a] + x))
/(c + d*x)]*Sqrt[-(((I*Sqrt[b]*d)/Sqrt[a] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*Ell
ipticF[I*ArcSinh[Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a]]/Sqrt[c + d*x]], (Sqrt[a]*c - I
*Sqrt[b]*d)/(Sqrt[a]*c + I*Sqrt[b]*d)]))/(a^2*d^2*Sqrt[-c - (I*Sqrt[b]*d)/Sqrt[a
]]*(c + d*x))))/(5*Sqrt[a + b/x^2]*x)

_______________________________________________________________________________________

Maple [B]  time = 0.15, size = 1145, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*((-a*b)^(1/2)*(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d
/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*E
llipticF((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^
(1/2)*d+a*c))^(1/2))*a*c^3*d+(-a*b)^(1/2)*(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2
)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*
b)^(1/2)*d-a*c))^(1/2)*EllipticF((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*
b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*b*c*d^3-3*(-(d*x+c)*a/((-a*b)^(1/2)
*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(
1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*EllipticF((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^
(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*a*b*c^2*d^2-3*b^2*(-(d
*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))
^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*EllipticF((-(d*x+c)*a/(
(-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*d
^4-(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*
d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*EllipticE((-(d*x
+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(
1/2))*a^2*c^4+2*(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-a*x+(-a*b)^(1/2))*d/(
(-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d-a*c))^(1/2)*Ell
ipticE((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1/2)*d-a*c)/((-a*b)^(1
/2)*d+a*c))^(1/2))*a*b*c^2*d^2+3*b^2*(-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2)*((-
a*x+(-a*b)^(1/2))*d/((-a*b)^(1/2)*d+a*c))^(1/2)*((a*x+(-a*b)^(1/2))*d/((-a*b)^(1
/2)*d-a*c))^(1/2)*EllipticE((-(d*x+c)*a/((-a*b)^(1/2)*d-a*c))^(1/2),(-((-a*b)^(1
/2)*d-a*c)/((-a*b)^(1/2)*d+a*c))^(1/2))*d^4+x^4*a^2*d^4+3*x^3*a^2*c*d^3+2*x^2*a^
2*c^2*d^2+x^2*a*b*d^4+3*x*a*b*c*d^3+2*a*b*c^2*d^2)/(d*x+c)^(1/2)/d^2/a^2/x/((a*x
^2+b)/x^2)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Exception raised: RuntimeError