3.631 \(\int \frac{1}{\sqrt{d+e x} \left (a-c x^2\right )^3} \, dx\)
Optimal. Leaf size=315 \[ -\frac{3 \left (-10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{3 \left (10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}-\frac{\sqrt{d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )^2}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )} \]
[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*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/
4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*d -
Sqrt[a]*e)^(5/2)) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*
d + Sqrt[a]*e)^(5/2))
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Rubi [A] time = 1.50032, antiderivative size = 315, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25
\[ -\frac{3 \left (-10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{c} d-\sqrt{a} e\right )^{5/2}}+\frac{3 \left (10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} \sqrt [4]{c} \left (\sqrt{a} e+\sqrt{c} d\right )^{5/2}}-\frac{\sqrt{d+e x} \left (a e \left (c d^2-7 a e^2\right )-6 c d x \left (c d^2-2 a e^2\right )\right )}{16 a^2 \left (a-c x^2\right ) \left (c d^2-a e^2\right )^2}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )} \]
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*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^(1/
4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*d -
Sqrt[a]*e)^(5/2)) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(c^
(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(1/4)*(Sqrt[c]*
d + Sqrt[a]*e)^(5/2))
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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 [A] time = 0.662082, size = 333, normalized size = 1.06 \[ -\frac{3 \left (-10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{32 a^{5/2} \left (\sqrt{c} d-\sqrt{a} e\right )^2 \sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{3 \left (10 \sqrt{a} \sqrt{c} d e+7 a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{32 a^{5/2} \left (\sqrt{a} e+\sqrt{c} d\right )^2 \sqrt{\sqrt{a} \sqrt{c} e+c d}}+\frac{\sqrt{d+e x} \left (11 a^3 e^3-a^2 c e \left (5 d^2+16 d e x+7 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+12 e^2 x^2\right )-6 c^3 d^3 x^3\right )}{16 a^2 \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[d + e*x]*(a - c*x^2)^3),x]
[Out]
(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)))/(16*a^2*(c*d^2 - a*e^2)^2*(a - c
*x^2)^2) - (3*(4*c*d^2 - 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2)*ArcTanh[(Sqrt[c]*Sqrt
[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/(32*a^(5/2)*(Sqrt[c]*d - Sqrt[a]*e)^2
*Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]) + (3*(4*c*d^2 + 10*Sqrt[a]*Sqrt[c]*d*e + 7*a*e^2
)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/(32*a^(5/2)*(S
qrt[c]*d + Sqrt[a]*e)^2*Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])
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Maple [B] time = 0.135, size = 950, normalized size = 3. \[ \text{result too large to display} \]
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]
-3/16*e/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)*(e*x+d)^
(3/2)*d+9/32*e/c/a^2*(a*c*e^2)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2-2*(a
*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)+3/16*e/a^2/(e*x+(a*c*e^2)^(1/2)/c)^2/(c*d-(a*c*e^
2)^(1/2))*(e*x+d)^(1/2)*d-11/32*e/c/a^2*(a*c*e^2)^(1/2)/(e*x+(a*c*e^2)^(1/2)/c)^
2/(c*d-(a*c*e^2)^(1/2))*(e*x+d)^(1/2)+21/32*e^3*c/a/(a*c*e^2)^(1/2)/(a*e^2+c*d^2
-2*(a*c*e^2)^(1/2)*d)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((
-c*d+(a*c*e^2)^(1/2))*c)^(1/2))+3/8*e*c^2/a^2/(a*c*e^2)^(1/2)/(a*e^2+c*d^2-2*(a*
c*e^2)^(1/2)*d)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(
a*c*e^2)^(1/2))*c)^(1/2))*d^2-15/16*e*c/a^2/(a*e^2+c*d^2-2*(a*c*e^2)^(1/2)*d)/((
-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)
^(1/2))*d-3/16*e/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)
*(e*x+d)^(3/2)*d-9/32*e/c/a^2*(a*c*e^2)^(1/2)/(e*x-(a*c*e^2)^(1/2)/c)^2/(a*e^2+c
*d^2+2*(a*c*e^2)^(1/2)*d)*(e*x+d)^(3/2)+3/16*e/a^2/(e*x-(a*c*e^2)^(1/2)/c)^2/(c*
d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)*d+11/32*e/c/a^2*(a*c*e^2)^(1/2)/(e*x-(a*c*e^2)^
(1/2)/c)^2/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2)+21/32*e^3*c/a/(a*c*e^2)^(1/2)/(a*
e^2+c*d^2+2*(a*c*e^2)^(1/2)*d)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)
^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+3/8*e*c^2/a^2/(a*c*e^2)^(1/2)/(a*e^2+c*d
^2+2*(a*c*e^2)^(1/2)*d)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/
((c*d+(a*c*e^2)^(1/2))*c)^(1/2))*d^2+15/16*e*c/a^2/(a*e^2+c*d^2+2*(a*c*e^2)^(1/2
)*d)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/
2))*c)^(1/2))*d
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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 = 1.65186, size = 7798, normalized size = 24.76 \[ \text{result too large to display} \]
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 + 10
5*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^1
8*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^1
4 - 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 + 1
0*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 - 5
292*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^1
0*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^1
0*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(2
7*(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 + 45*a^7*c^9*d^16*e^4 - 1
20*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^1
8 + 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 - 1
69*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 + 240
1*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 + 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))) - 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)
_______________________________________________________________________________________
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
_______________________________________________________________________________________
GIAC/XCAS [A] time = 175.189, 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