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

Optimal. Leaf size=494 \[ -\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{693 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{693 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{693 e^5}+\frac{20 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{693 e^3}+\frac{2 \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]

[Out]

(8*Sqrt[d + e*x]*(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*a^2*e^4 - 24*c*d*e*(c*d^2 + 2
*a*e^2)*x)*Sqrt[a + c*x^2])/(693*e^5) + (20*Sqrt[d + e*x]*(8*c*d^2 + 9*a*e^2 - 7
*c*d*e*x)*(a + c*x^2)^(3/2))/(693*e^3) + (2*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(11
*e) + (16*Sqrt[-a]*Sqrt[c]*d*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*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)])/(693*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqr
t[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-a]*(c*d^2 + a*e^2)*(32*c^2*d^
4 + 69*a*c*d^2*e^2 + 45*a^2*e^4)*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)])/(693*Sqrt[c]*e^6*Sqrt[d + e*x]*Sqrt[a +
c*x^2])

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Rubi [A]  time = 1.3612, antiderivative size = 494, 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 \[ -\frac{16 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{693 \sqrt{c} e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{693 e^6 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{8 \sqrt{a+c x^2} \sqrt{d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{693 e^5}+\frac{20 \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{693 e^3}+\frac{2 \left (a+c x^2\right )^{5/2} \sqrt{d+e x}}{11 e} \]

Antiderivative was successfully verified.

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

[Out]

(8*Sqrt[d + e*x]*(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*a^2*e^4 - 24*c*d*e*(c*d^2 + 2
*a*e^2)*x)*Sqrt[a + c*x^2])/(693*e^5) + (20*Sqrt[d + e*x]*(8*c*d^2 + 9*a*e^2 - 7
*c*d*e*x)*(a + c*x^2)^(3/2))/(693*e^3) + (2*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(11
*e) + (16*Sqrt[-a]*Sqrt[c]*d*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*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)])/(693*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqr
t[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (16*Sqrt[-a]*(c*d^2 + a*e^2)*(32*c^2*d^
4 + 69*a*c*d^2*e^2 + 45*a^2*e^4)*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)])/(693*Sqrt[c]*e^6*Sqrt[d + e*x]*Sqrt[a +
c*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 6.48943, size = 634, normalized size = 1.28 \[ \frac{2 \sqrt{d+e x} \left (-\frac{8 d e^2 \left (a+c x^2\right ) \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right )}{d+e x}-8 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right ) \left (333 a^2 e^4+2 a c e^2 \left (178 d^2-131 d e x+108 e^2 x^2\right )+c^2 \left (128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right )\right )+\frac{8 \sqrt{a} e \sqrt{d+e x} \left (21 i a^{3/2} c d^2 e^3+45 i a^{5/2} e^5+93 a^2 \sqrt{c} d e^4+93 a c^{3/2} d^3 e^2+8 i \sqrt{a} c^2 d^4 e+32 c^{5/2} d^5\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{693 e^7 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*((-8*d*e^2*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*(a + c*x^
2))/(d + e*x) + e^2*(a + c*x^2)*(333*a^2*e^4 + 2*a*c*e^2*(178*d^2 - 131*d*e*x +
108*e^2*x^2) + c^2*(128*d^4 - 96*d^3*e*x + 80*d^2*e^2*x^2 - 70*d*e^3*x^3 + 63*e^
4*x^4)) - (8*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32*c^2*d^4 + 93*a*c*d^2*e^
2 + 93*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]
*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sq
rt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[
a]*e)] + (8*Sqrt[a]*e*(32*c^(5/2)*d^5 + (8*I)*Sqrt[a]*c^2*d^4*e + 93*a*c^(3/2)*d
^3*e^2 + (21*I)*a^(3/2)*c*d^2*e^3 + 93*a^2*Sqrt[c]*d*e^4 + (45*I)*a^(5/2)*e^5)*S
qrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x
)/(d + e*x))]*Sqrt[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 -
(I*Sqrt[a]*e)/Sqrt[c]]))/(693*e^7*Sqrt[a + c*x^2])

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Maple [B]  time = 0.036, size = 1970, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)/(e*x+d)^(1/2),x)

[Out]

2/693*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)*(256*(-(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^4*d^7+333*a^3*c*d*e^6+356*a^2*c^2*
d^3*e^4+128*a*c^3*d^5*e^2-7*x^6*c^4*d*e^6+279*x^5*a*c^3*e^7+10*x^5*c^4*d^2*e^5-1
6*x^4*c^4*d^3*e^4+549*x^3*a^2*c^2*e^7+32*x^3*c^4*d^4*e^3+128*x^2*c^4*d^5*e^2+333
*x*a^3*c*e^7+340*x^2*a*c^3*d^3*e^4+94*x*a^2*c^2*d^2*e^5+32*x*a*c^3*d^4*e^3-53*x^
4*a*c^3*d*e^6-576*(-(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)*E
llipticF((-(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^2*d^3*e^4-360*(-(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^3*e^7+104*x^3*a*c^3*d
^2*e^5+287*x^2*a^2*c^2*d*e^6-384*(-(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^3*c*d*e^6+1000*(-(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^3*d^5*e^2+744*(-(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^3
*c*d*e^6+1488*(-(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)*Ellip
ticE((-(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^2*d^3*e^4-192*(-(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^3*d^5*e^2-256*(-(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^3*d^
6*e-912*(-(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^2*c*d^2*e^5-808*(-(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*c^2*d^4*e^3+63*x^7*c
^4*e^7)/c/e^7/(c*e*x^3+c*d*x^2+a*e*x+a*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)/sqrt(e*x + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(5/2)/(e*x+d)**(1/2),x)

[Out]

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

[Out]

Exception raised: RuntimeError