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

Optimal. Leaf size=321 \[ \frac{\sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

[Out]

-(((a*e - c*d*x)*Sqrt[d + e*x])/(a*c*Sqrt[a + c*x^2])) - (d*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]) + ((c*d^2 + a*e^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]*c^(3/2)*
Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.712366, antiderivative size = 321, 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 \[ \frac{\sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a e-c d x)}{a c \sqrt{a+c x^2}}-\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

Antiderivative was successfully verified.

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

[Out]

-(((a*e - c*d*x)*Sqrt[d + e*x])/(a*c*Sqrt[a + c*x^2])) - (d*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]) + ((c*d^2 + a*e^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]*c^(3/2)*
Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 120.31, size = 296, normalized size = 0.92 \[ - \frac{d \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{\sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{c^{\frac{3}{2}} \sqrt{- a} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{\sqrt{d + e x} \left (a e - c d x\right )}{a c \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*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*e**2 + c*d**2)
*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sq
rt(-a)))/(c**(3/2)*sqrt(-a)*sqrt(a + c*x**2)*sqrt(d + e*x)) - sqrt(d + e*x)*(a*e
- c*d*x)/(a*c*sqrt(a + c*x**2))

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Mathematica [C]  time = 7.92517, size = 542, normalized size = 1.69 \[ \frac{\sqrt{d+e x} (c d x-a e)}{a c \sqrt{a+c x^2}}-\frac{(d+e x)^{3/2} \left (d \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (\frac{a e^2}{(d+e x)^2}+c \left (\frac{d}{d+e x}-1\right )^2\right )-\frac{\sqrt{a} e \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{-\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} \sqrt{\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}+\frac{\sqrt{c} d \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{-\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} \sqrt{\frac{i \sqrt{a} e}{\sqrt{c} (d+e x)}-\frac{d}{d+e x}+1} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}\right )}{a c e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{a+\frac{c (d+e x)^2 \left (\frac{d}{d+e x}-1\right )^2}{e^2}}} \]

Antiderivative was successfully verified.

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

[Out]

((-(a*e) + c*d*x)*Sqrt[d + e*x])/(a*c*Sqrt[a + c*x^2]) - ((d + e*x)^(3/2)*(d*Sqr
t[-d - (I*Sqrt[a]*e)/Sqrt[c]]*((a*e^2)/(d + e*x)^2 + c*(-1 + d/(d + e*x))^2) + (
Sqrt[c]*d*((-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]*e*(Sqrt[c]*d + I*
Sqrt[a]*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 = 0.058, size = 685, normalized size = 2.1 \[{\frac{1}{e \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ) a{c}^{2}} \left ( -\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}{\it EllipticF} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) \sqrt{-ac}a{e}^{3}-\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}{\it EllipticF} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) \sqrt{-ac}c{d}^{2}e+\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}{\it EllipticE} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) acd{e}^{2}+\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}{\it EllipticE} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ){c}^{2}{d}^{3}+{x}^{2}{c}^{2}d{e}^{2}-xac{e}^{3}+x{c}^{2}{d}^{2}e-ad{e}^{2}c \right ) \sqrt{ex+d}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)/(c*x^2+a)^(3/2),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*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. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

[Out]

Exception raised: RuntimeError