3.636 \(\int \frac{1}{\sqrt{d+e x} \left (a+c x^2\right )^3} \, dx\)

Optimal. Leaf size=920 \[ \frac{\sqrt{d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (c x^2+a\right )} \]

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*(c*d^2 + a*e^2)*(a + c*x^2)^2) + (Sqrt[d + e*
x]*(a*e*(c*d^2 + 7*a*e^2) + 6*c*d*(c*d^2 + 2*a*e^2)*x))/(16*a^2*(c*d^2 + a*e^2)^
2*(a + c*x^2)) + (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 + 2*Sqrt[c]*d*Sqrt[
c*d^2 + a*e^2]*(c*d^2 + 2*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*Sq
rt[2]*a^2*c^(1/4)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) -
(3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 + 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(
c*d^2 + 2*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^(1/4
)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^2*d^4
+ 5*a*c*d^2*e^2 + 7*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 2*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^(1/4)*(c*d^2 + a*e^2)^(5
/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7
*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 2*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] + Sq
rt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(1/4)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d +
Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 14.4306, antiderivative size = 920, 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 \[ \frac{\sqrt{d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (c x^2+a\right )} \]

Antiderivative was successfully verified.

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

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*(c*d^2 + a*e^2)*(a + c*x^2)^2) + (Sqrt[d + e*
x]*(a*e*(c*d^2 + 7*a*e^2) + 6*c*d*(c*d^2 + 2*a*e^2)*x))/(16*a^2*(c*d^2 + a*e^2)^
2*(a + c*x^2)) + (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 + 2*Sqrt[c]*d*Sqrt[
c*d^2 + a*e^2]*(c*d^2 + 2*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*Sq
rt[2]*a^2*c^(1/4)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) -
(3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 + 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(
c*d^2 + 2*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^(1/4
)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^2*d^4
+ 5*a*c*d^2*e^2 + 7*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 2*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^(1/4)*(c*d^2 + a*e^2)^(5
/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7
*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 2*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] + Sq
rt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(1/4)*(c*d^2 + a*e^2)^(5/2)*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(1/(c*x**2+a)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.75386, size = 348, normalized size = 0.38 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (11 a^3 e^3+a^2 c e \left (5 d^2+16 d e x+7 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+12 e^2 x^2\right )+6 c^3 d^3 x^3\right )}{\left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}-\frac{3 i \left (-10 i \sqrt{a} \sqrt{c} d e-7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\left (\sqrt{c} d-i \sqrt{a} e\right )^2 \sqrt{c d-i \sqrt{a} \sqrt{c} e}}+\frac{3 i \left (10 i \sqrt{a} \sqrt{c} d e-7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\left (\sqrt{c} d+i \sqrt{a} e\right )^2 \sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{32 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*Sqrt[d + e*x]*(11*a^3*e^3 + 6*c^3*d^3*x^3 + a^2*c*e*(5*d^2 + 16*d*e*
x + 7*e^2*x^2) + a*c^2*d*x*(10*d^2 + d*e*x + 12*e^2*x^2)))/((c*d^2 + a*e^2)^2*(a
+ c*x^2)^2) - ((3*I)*(4*c*d^2 - (10*I)*Sqrt[a]*Sqrt[c]*d*e - 7*a*e^2)*ArcTanh[(
Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]])/((Sqrt[c]*d - I*Sqrt[a]
*e)^2*Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]) + ((3*I)*(4*c*d^2 + (10*I)*Sqrt[a]*Sqrt[c
]*d*e - 7*a*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]
])/((Sqrt[c]*d + I*Sqrt[a]*e)^2*Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]))/(32*a^(5/2))

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Maple [F]  time = 2.295, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}}{\frac{1}{\sqrt{ex+d}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^3*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64551, size = 7802, normalized size = 8.48 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^3*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

1/64*(3*(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*
e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sq
rt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 1
05*a^4*d*e^8 + (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c
^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d
^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11
*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*
a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4
*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^
5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d
^2*e^8 + a^10*e^10))*log(27*(336*c^4*d^8*e^5 + 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2
*d^4*e^9 + 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) + 27*(42*a^3*c^4*d
^8*e^6 + 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 + 623*a^6*c*d^2*e^12 + 343*a
^7*e^14 + (4*a^5*c^8*d^15 + 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 + 205*a^8
*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 + 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^1
2 + 11*a^12*c*d*e^14)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c
^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*
d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 2
52*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*
c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt(-(16*c^4*d^9 + 84*a*c^
3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d
^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*
e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^
4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*
e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^
10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d
^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^
2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))) - 3
*(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a
^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt(-(16
*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*
d*e^8 + (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*
e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12
+ 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 +
10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7
*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^
14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10
+ 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8
+ a^10*e^10))*log(27*(336*c^4*d^8*e^5 + 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^
9 + 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) - 27*(42*a^3*c^4*d^8*e^6
+ 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 + 623*a^6*c*d^2*e^12 + 343*a^7*e^14
+ (4*a^5*c^8*d^15 + 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 + 205*a^8*c^5*d^
9*e^6 + 240*a^9*c^4*d^7*e^8 + 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 + 11*
a^12*c*d*e^14)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*
e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^
2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10
*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4
*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e
^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 + 5
*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a
^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14
+ 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 4
5*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*
d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16
+ 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*
a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))) + 3*(a^4*c
^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*
e^4)*x^4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt(-(16*c^4*d^
9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 -
(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5
*a^9*c*d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974
*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6
*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e
^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45
*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6
*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*
e^10))*log(27*(336*c^4*d^8*e^5 + 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 + 480
2*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) + 27*(42*a^3*c^4*d^8*e^6 + 213*a
^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 + 623*a^6*c*d^2*e^12 + 343*a^7*e^14 - (4*a
^5*c^8*d^15 + 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 + 205*a^8*c^5*d^9*e^6 +
240*a^9*c^4*d^7*e^8 + 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 + 11*a^12*c*
d*e^14)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 +
5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*
a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^
10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 +
10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 18
9*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d^10 + 5*a^6*c^
4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^1
0)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*
a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c
^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^
10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a
^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3
*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))) - 3*(a^4*c^2*d^4
+ 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^
4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt(-(16*c^4*d^9 + 84*
a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c
^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*
d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^
2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d
^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 25
2*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c
^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^
8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))*
log(27*(336*c^4*d^8*e^5 + 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 + 4802*a^3*c
*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) - 27*(42*a^3*c^4*d^8*e^6 + 213*a^4*c^3*
d^6*e^8 + 515*a^5*c^2*d^4*e^10 + 623*a^6*c*d^2*e^12 + 343*a^7*e^14 - (4*a^5*c^8*
d^15 + 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 + 205*a^8*c^5*d^9*e^6 + 240*a^
9*c^4*d^7*e^8 + 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 + 11*a^12*c*d*e^14)
*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^
3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9
*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10
+ 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^1
4*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c
^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e
^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10)*sqrt
(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d
^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16
*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 21
0*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2
*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^
4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))) + 4*(5*a^2*c*d^2*e + 11*
a^3*e^3 + 6*(c^3*d^3 + 2*a*c^2*d*e^2)*x^3 + (a*c^2*d^2*e + 7*a^2*c*e^3)*x^2 + 2*
(5*a*c^2*d^3 + 8*a^2*c*d*e^2)*x)*sqrt(e*x + d))/(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 +
a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4
+ 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*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(1/(c*x**2+a)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 82.1734, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^3*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Done