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

Optimal. Leaf size=849 \[ \frac{\sqrt{d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )} \]

[Out]

(x*Sqrt[d + e*x])/(4*a*(a + c*x^2)^2) + (Sqrt[d + e*x]*(a*d*e + (6*c*d^2 + 5*a*e
^2)*x))/(16*a^2*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d
*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c
*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*
e^2]]])/(32*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^
2 + a*e^2]]) - (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*
d^2 + 5*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)
*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(3/4)*
(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^(3/2)*d^3
+ 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*Log[Sqrt[c*d^2 +
a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +
Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d
+ Sqrt[c*d^2 + a*e^2]]) + (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 +
a*e^2]*(6*c*d^2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[
c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*
c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 6.65998, antiderivative size = 849, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ \frac{\sqrt{d+e x} x}{4 a \left (c x^2+a\right )^2}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (6 c^{3/2} d^3+8 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (6 c d^2+5 a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{3/4} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a d e+\left (6 c d^2+5 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right ) \left (c x^2+a\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[d + e*x])/(4*a*(a + c*x^2)^2) + (Sqrt[d + e*x]*(a*d*e + (6*c*d^2 + 5*a*e
^2)*x))/(16*a^2*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d
*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c
*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*
e^2]]])/(32*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^
2 + a*e^2]]) - (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(6*c*
d^2 + 5*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)
*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(3/4)*
(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(6*c^(3/2)*d^3
+ 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(6*c*d^2 + 5*a*e^2))*Log[Sqrt[c*d^2 +
a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +
Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d
+ Sqrt[c*d^2 + a*e^2]]) + (e*(6*c^(3/2)*d^3 + 8*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 +
a*e^2]*(6*c*d^2 + 5*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[
c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*
c^(3/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^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((e*x+d)**(1/2)/(c*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [C]  time = 0.583046, size = 318, normalized size = 0.37 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (\frac{\left (a+c x^2\right ) \left (a e (d+5 e x)+6 c d^2 x\right )}{a e^2+c d^2}+4 a x\right )}{\left (a+c x^2\right )^2}-\frac{i \left (-18 i \sqrt{a} \sqrt{c} d e-5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c} \left (\sqrt{c} d-i \sqrt{a} e\right ) \sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{i \left (18 i \sqrt{a} \sqrt{c} d e-5 a e^2+12 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{32 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*Sqrt[d + e*x]*(4*a*x + ((a + c*x^2)*(6*c*d^2*x + a*e*(d + 5*e*x)))/(
c*d^2 + a*e^2)))/(a + c*x^2)^2 - (I*(12*c*d^2 - (18*I)*Sqrt[a]*Sqrt[c]*d*e - 5*a
*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]])/(Sqrt[c]
*(Sqrt[c]*d - I*Sqrt[a]*e)*Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]) + (I*(12*c*d^2 + (18
*I)*Sqrt[a]*Sqrt[c]*d*e - 5*a*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + I*
Sqrt[a]*Sqrt[c]*e]])/(Sqrt[c]*(Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[c*d + I*Sqrt[a]*Sqr
t[c]*e]))/(32*a^(5/2))

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Maple [F]  time = 2.823, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}}\sqrt{ex+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.473361, size = 5090, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*((a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 +
a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105
*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*s
qrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6
*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6
*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c
^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c
*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) + (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*
e^8 + 200*a^5*c*d*e^10 - (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*
e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e^10)*sqrt(-(441*c^2*
d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2
+ 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*
e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e
^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*
c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12
+ 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4
*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 +
3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) - (a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^
2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5
*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*
a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a
^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^
6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6
+ 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7
884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) - (126*a^3*
c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 - (12*a^5*c^7*d^10 + 55*a^6
*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*
a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*
c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9
*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d^7 + 420*a
*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + (a^5*c^4*d^6 + 3*a^6*c^3*d^4*
e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12
+ 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^
8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5
*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) + (a^4*c*d^2 + a
^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)*sqrt
(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^
4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^
10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a
^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 +
a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e
^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 + 625*a^3*e
^11)*sqrt(e*x + d) + (126*a^3*c^3*d^5*e^6 + 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^
10 + (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 + 84*a^8*c^4*d^4
*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e^10)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*
d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4
+ 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12
)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3*e^4 + 105*a^3*d*e^6 -
(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6)*sqrt(-(441*c^
2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8*d^10*e^
2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^10*c^4*d^
2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 +
a^8*c*e^6))) - (a^4*c*d^2 + a^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*
c^2*d^2 + a^4*c*e^2)*x^2)*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3
*e^4 + 105*a^3*d*e^6 - (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^
8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^
12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d
^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2
+ 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))*log((3024*c^3*d^6*e^5 + 7884*a*c^2*d^4*e^7 +
5625*a^2*c*d^2*e^9 + 625*a^3*e^11)*sqrt(e*x + d) - (126*a^3*c^3*d^5*e^6 + 318*a^
4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 + (12*a^5*c^7*d^10 + 55*a^6*c^6*d^8*e^2 + 98*a^
7*c^5*d^6*e^4 + 84*a^8*c^4*d^4*e^6 + 34*a^9*c^3*d^2*e^8 + 5*a^10*c^2*e^10)*sqrt(
-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 + 6*a^6*c^8
*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 + 6*a^1
0*c^4*d^2*e^10 + a^11*c^3*e^12)))*sqrt(-(144*c^3*d^7 + 420*a*c^2*d^5*e^2 + 385*a
^2*c*d^3*e^4 + 105*a^3*d*e^6 - (a^5*c^4*d^6 + 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*
e^4 + a^8*c*e^6)*sqrt(-(441*c^2*d^4*e^10 + 1050*a*c*d^2*e^12 + 625*a^2*e^14)/(a^
5*c^9*d^12 + 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 + 20*a^8*c^6*d^6*e^6 + 15*a
^9*c^5*d^4*e^8 + 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4*d^6 + 3*a^6*c^3
*d^4*e^2 + 3*a^7*c^2*d^2*e^4 + a^8*c*e^6))) + 4*(a*c*d*e*x^2 + a^2*d*e + (6*c^2*
d^2 + 5*a*c*e^2)*x^3 + (10*a*c*d^2 + 9*a^2*e^2)*x)*sqrt(e*x + d))/(a^4*c*d^2 + a
^5*e^2 + (a^2*c^3*d^2 + a^3*c^2*e^2)*x^4 + 2*(a^3*c^2*d^2 + a^4*c*e^2)*x^2)

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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)**(1/2)/(c*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out