3.684 \(\int \frac{(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=368 \[ \frac{\sqrt{d+e x} (a e+4 c d x)}{6 a^2 c \sqrt{a+c x^2}}-\frac{\sqrt{\frac{c x^2}{a}+1} \left (a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(3*a*c*(a + c*x^2)^(3/2)) + ((a*e + 4*c*d*x)*Sqrt
[d + e*x])/(6*a^2*c*Sqrt[a + c*x^2]) + (2*d*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*El
lipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[
c]*d - a*e)])/(3*(-a)^(3/2)*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-
a]*e)]*Sqrt[a + c*x^2]) - ((4*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)])/(6*(-a)^(3/2)*c^(3/2)*Sqrt[d
+ e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.03905, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{\sqrt{d+e x} (a e+4 c d x)}{6 a^2 c \sqrt{a+c x^2}}-\frac{\sqrt{\frac{c x^2}{a}+1} \left (a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}+\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)/(a + c*x^2)^(5/2),x]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(3*a*c*(a + c*x^2)^(3/2)) + ((a*e + 4*c*d*x)*Sqrt
[d + e*x])/(6*a^2*c*Sqrt[a + c*x^2]) + (2*d*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*El
lipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[
c]*d - a*e)])/(3*(-a)^(3/2)*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-
a]*e)]*Sqrt[a + c*x^2]) - ((4*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)])/(6*(-a)^(3/2)*c^(3/2)*Sqrt[d
+ e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 164.87, size = 342, normalized size = 0.93 \[ \frac{2 d \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 \sqrt{c} \left (- a\right )^{\frac{3}{2}} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} - \frac{\sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + 4 c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{6 c^{\frac{3}{2}} \left (- a\right )^{\frac{3}{2}} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{\sqrt{d + e x} \left (a e - c d x\right )}{3 a c \left (a + c x^{2}\right )^{\frac{3}{2}}} + \frac{\sqrt{d + e x} \left (\frac{a e}{2} + 2 c d x\right )}{3 a^{2} c \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)/(c*x**2+a)**(5/2),x)

[Out]

2*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)))/(3*sqrt(c)*(-a)**(3/2)*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 + 4
*c*d**2)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt
(c)*d*sqrt(-a)))/(6*c**(3/2)*(-a)**(3/2)*sqrt(a + c*x**2)*sqrt(d + e*x)) - sqrt(
d + e*x)*(a*e - c*d*x)/(3*a*c*(a + c*x**2)**(3/2)) + sqrt(d + e*x)*(a*e/2 + 2*c*
d*x)/(3*a**2*c*sqrt(a + c*x**2))

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Mathematica [C]  time = 6.64951, size = 504, normalized size = 1.37 \[ \frac{\sqrt{d+e x} \left (\frac{-2 a^2 e+2 a c x (6 d+e x)+8 c^2 d x^3}{a^2 c \left (a+c x^2\right )}+\frac{(d+e x) \left (-\frac{8 d e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{(d+e x)^2}+\frac{2 \sqrt{a} e \left (4 \sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}+\frac{8 i \sqrt{c} d \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{d+e x}}\right )}{a^2 c e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{12 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(3/2)/(a + c*x^2)^(5/2),x]

[Out]

(Sqrt[d + e*x]*((-2*a^2*e + 8*c^2*d*x^3 + 2*a*c*x*(6*d + e*x))/(a^2*c*(a + c*x^2
)) + ((d + e*x)*((-8*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2))/(d + e*
x)^2 + ((8*I)*Sqrt[c]*d*(Sqrt[c]*d + I*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))]*EllipticE[I*Arc
Sinh[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[d + e*x] + (2*Sqrt[a]*e*(4*Sqrt[c]*d + I*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))]*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*Sqrt[a]*e)])/Sqrt[d + e*x]))/(
a^2*c*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])))/(12*Sqrt[a + c*x^2])

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Maple [B]  time = 0.072, size = 1633, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)/(c*x^2+a)^(5/2),x)

[Out]

-1/6*(3*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))*x^2*a*c^2*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)+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))*x^2*a*c*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)*(-a*c)^(1/2)+4*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))*x
^2*c^2*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)*(-a*c)
^(1/2)-4*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))*x^2*a*c^2*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)-4*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))*x^2*c^3*d^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)+3*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^2*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)+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^2*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)*(-a*c)^
(1/2)+4*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^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)*(-a*c)^(1/2)-4*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^2*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)-4*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^2
*d^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)-4*x^4*c^3*d*e^
2-x^3*a*c^2*e^3-4*x^3*c^3*d^2*e-7*x^2*a*c^2*d*e^2+x*a^2*c*e^3-6*x*a*c^2*d^2*e+a^
2*c*d*e^2)/(e*x+d)^(1/2)/a^2/e/c^2/(c*x^2+a)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(c*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/(c*x^2 + a)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(c*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

integral((e*x + d)^(3/2)/((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)), 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)**(5/2),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)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError