3.629 \(\int \frac{(d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx\)
Optimal. Leaf size=268 \[ -\frac{3 \left (-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{3 \left (2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{5/4} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\sqrt{d+e x} (a e-6 c d x)}{16 a^2 c \left (a-c x^2\right )}+\frac{\sqrt{d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]
[Out]
((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) - ((a*e - 6*c*d*x)*Sqrt[d +
e*x])/(16*a^2*c*(a - c*x^2)) - (3*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcT
anh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(5/4)*Sq
rt[Sqrt[c]*d - Sqrt[a]*e]) + (3*(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan
h[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(5/4)*Sqrt
[Sqrt[c]*d + Sqrt[a]*e])
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Rubi [A] time = 1.19399, antiderivative size = 268, 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
\[ -\frac{3 \left (-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{5/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{3 \left (2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{5/4} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\sqrt{d+e x} (a e-6 c d x)}{16 a^2 c \left (a-c x^2\right )}+\frac{\sqrt{d+e x} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/(a - c*x^2)^3,x]
[Out]
((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) - ((a*e - 6*c*d*x)*Sqrt[d +
e*x])/(16*a^2*c*(a - c*x^2)) - (3*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcT
anh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(5/4)*Sq
rt[Sqrt[c]*d - Sqrt[a]*e]) + (3*(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan
h[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(5/4)*Sqrt
[Sqrt[c]*d + Sqrt[a]*e])
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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)**(3/2)/(-c*x**2+a)**3,x)
[Out]
Timed out
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Mathematica [A] time = 0.654982, size = 238, normalized size = 0.89 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (3 a^2 e+a c x (10 d+e x)-6 c^2 d x^3\right )}{\left (a-c x^2\right )^2}-\frac{3 \left (-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{3 \left (2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}}{32 a^{5/2} c} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/(a - c*x^2)^3,x]
[Out]
((2*Sqrt[a]*Sqrt[d + e*x]*(3*a^2*e - 6*c^2*d*x^3 + a*c*x*(10*d + e*x)))/(a - c*x
^2)^2 - (3*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e
*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e] + (3*(4*c*d^2
+ 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqr
t[a]*Sqrt[c]*e]])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c)
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Maple [B] time = 0.06, size = 778, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(-c*x^2+a)^3,x)
[Out]
-3/8*e/(c*e^2*x^2-a*e^2)^2*c*d/a^2*(e*x+d)^(7/2)+1/16*e^3/(c*e^2*x^2-a*e^2)^2/a*
(e*x+d)^(5/2)+9/8*e/(c*e^2*x^2-a*e^2)^2/a^2*(e*x+d)^(5/2)*c*d^2+1/2*e^3/(c*e^2*x
^2-a*e^2)^2*d/a*(e*x+d)^(3/2)-9/8*e/(c*e^2*x^2-a*e^2)^2*d^3/a^2*(e*x+d)^(3/2)*c+
3/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)-9/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x
+d)^(1/2)*d^2+3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(1/2)*d^4-3/32*e^9*a^2*c/(
a^5*c^3*e^10)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2)*arctanh(e^2*a*c*(
e*x+d)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2))+3/8*e^7*a*c^2/(a^5*c^3*
e^10)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2)*arctanh(e^2*a*c*(e*x+d)^(
1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2))*d^2+3/16*e^3/a/(a^2*d*e^4*c^2+(
a^5*c^3*e^10)^(1/2))^(1/2)*arctanh(e^2*a*c*(e*x+d)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3
*e^10)^(1/2))^(1/2))*d-3/32*e^9*a^2*c/(a^5*c^3*e^10)^(1/2)/(-a^2*d*e^4*c^2+(a^5*
c^3*e^10)^(1/2))^(1/2)*arctan(e^2*a*c*(e*x+d)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^1
0)^(1/2))^(1/2))+3/8*e^7*a*c^2/(a^5*c^3*e^10)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^1
0)^(1/2))^(1/2)*arctan(e^2*a*c*(e*x+d)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2
))^(1/2))*d^2-3/16*e^3/a/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2)*arctan(e^2*
a*c*(e*x+d)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2))*d
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} - a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x + d)^(3/2)/(c*x^2 - a)^3,x, algorithm="maxima")
[Out]
-integrate((e*x + d)^(3/2)/(c*x^2 - a)^3, x)
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Fricas [A] time = 0.26172, size = 2367, normalized size = 8.83 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(e*x + d)^(3/2)/(c*x^2 - a)^3,x, algorithm="fricas")
[Out]
1/64*(3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2
+ 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d
^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 - 12
*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^2*e^6 - a^4*c*e^8 - (4*a
^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6
*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 +
(a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5
*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*
sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sq
rt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2
*e^2))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 27*(2*
a^3*c^2*d^2*e^6 - a^4*c*e^8 - (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e
^4)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt((16*c^2*d^5
- 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7
*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))) + 3*(a^2
*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^
4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7
*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7
+ a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (4*a^5*c^6*d^5 -
7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2
+ a^7*c^5*e^4)))*sqrt((16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2
- a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5
*c^3*d^2 - a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*c^2
*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5
*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))*log(2
7*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 27*(2*a^3*c^2*d^2*
e^6 - a^4*c*e^8 + (4*a^5*c^6*d^5 - 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(e^1
0/(a^5*c^7*d^4 - 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt((16*c^2*d^5 - 20*a*c*d^
3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 - a^6*c^2*e^2)*sqrt(e^10/(a^5*c^7*d^4 - 2*a^6
*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 - a^6*c^2*e^2))) - 4*(6*c^2*d*x^3 - a
*c*e*x^2 - 10*a*c*d*x - 3*a^2*e)*sqrt(e*x + d))/(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a
^4*c)
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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)**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(-(e*x + d)^(3/2)/(c*x^2 - a)^3,x, algorithm="giac")
[Out]
Timed out