3.614 \(\int \frac{(d+e x)^{5/2}}{\left (a-c x^2\right )^2} \, dx\)
Optimal. Leaf size=231 \[ -\frac{\left (3 \sqrt{a} e+2 \sqrt{c} d\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac{\left (2 \sqrt{c} d-3 \sqrt{a} e\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{7/4}}+\frac{(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac{d e \sqrt{d+e x}}{2 a c} \]
[Out]
(d*e*Sqrt[d + e*x])/(2*a*c) + ((a*e + c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a - c*x^2)
) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(2*Sqrt[c]*d + 3*Sqrt[a]*e)*ArcTanh[(c^(1/4)*
Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(7/4)) + ((2*Sqrt[c]*d
- 3*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sq
rt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(7/4))
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Rubi [A] time = 1.01258, antiderivative size = 231, 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
\[ -\frac{\left (3 \sqrt{a} e+2 \sqrt{c} d\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{4 a^{3/2} c^{7/4}}+\frac{\left (2 \sqrt{c} d-3 \sqrt{a} e\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{4 a^{3/2} c^{7/4}}+\frac{(d+e x)^{3/2} (a e+c d x)}{2 a c \left (a-c x^2\right )}+\frac{d e \sqrt{d+e x}}{2 a c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)/(a - c*x^2)^2,x]
[Out]
(d*e*Sqrt[d + e*x])/(2*a*c) + ((a*e + c*d*x)*(d + e*x)^(3/2))/(2*a*c*(a - c*x^2)
) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(2*Sqrt[c]*d + 3*Sqrt[a]*e)*ArcTanh[(c^(1/4)*
Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(7/4)) + ((2*Sqrt[c]*d
- 3*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sq
rt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*c^(7/4))
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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)**(5/2)/(-c*x**2+a)**2,x)
[Out]
Timed out
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Mathematica [A] time = 0.445088, size = 254, normalized size = 1.1 \[ \frac{-\frac{2 \sqrt{a} c \sqrt{d+e x} \left (a e (2 d+e x)+c d^2 x\right )}{c x^2-a}-\frac{\sqrt{c} \left (3 \sqrt{a} e+2 \sqrt{c} d\right ) \left (\sqrt{c} d-\sqrt{a} e\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\sqrt{c} \left (2 \sqrt{c} d-3 \sqrt{a} e\right ) \left (\sqrt{a} e+\sqrt{c} d\right )^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}}{4 a^{3/2} c^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)/(a - c*x^2)^2,x]
[Out]
((-2*Sqrt[a]*c*Sqrt[d + e*x]*(c*d^2*x + a*e*(2*d + e*x)))/(-a + c*x^2) - (Sqrt[c
]*(Sqrt[c]*d - Sqrt[a]*e)^2*(2*Sqrt[c]*d + 3*Sqrt[a]*e)*ArcTanh[(Sqrt[c]*Sqrt[d
+ e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e] + (Sqrt[c]
*(2*Sqrt[c]*d - 3*Sqrt[a]*e)*(Sqrt[c]*d + Sqrt[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])/(4*a^(3/2)
*c^2)
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Maple [B] time = 0.098, size = 779, normalized size = 3.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)/(-c*x^2+a)^2,x)
[Out]
-1/2*e^3/(c*e^2*x^2-a*e^2)/c*(e*x+d)^(3/2)-1/2*e/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(3/
2)*d^2-1/2*e^3/(c*e^2*x^2-a*e^2)*d/c*(e*x+d)^(1/2)+1/2*e/(c*e^2*x^2-a*e^2)*d^3/a
*(e*x+d)^(1/2)-e^6*a^2*c/(a^3*c^3*e^6)^(1/2)/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*
a)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2)
)*d+1/2*e^4*a*c^2/(a^3*c^3*e^6)^(1/2)/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2
)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))*d^3-3
/4*e^4*a/c/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/
2)/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))+1/4*e^2/((a*c^2*e^2*d+(a^3*c^3*e
^6)^(1/2))*a)^(1/2)*arctanh(a*c*e*(e*x+d)^(1/2)/((a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2
))*a)^(1/2))*d^2-e^6*a^2*c/(a^3*c^3*e^6)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2
))*a)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1
/2))*d+1/2*e^4*a*c^2/(a^3*c^3*e^6)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^
(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))*d
^3+3/4*e^4*a/c/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2)*arctan(a*c*e*(e*x+d)
^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6)^(1/2))*a)^(1/2))-1/4*e^2/((-a*c^2*e^2*d+(a^3
*c^3*e^6)^(1/2))*a)^(1/2)*arctan(a*c*e*(e*x+d)^(1/2)/((-a*c^2*e^2*d+(a^3*c^3*e^6
)^(1/2))*a)^(1/2))*d^2
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} - a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(c*x^2 - a)^2,x, algorithm="maxima")
[Out]
integrate((e*x + d)^(5/2)/(c*x^2 - a)^2, x)
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Fricas [A] time = 0.265428, size = 1850, normalized size = 8.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(c*x^2 - a)^2,x, algorithm="fricas")
[Out]
-1/8*((a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*
c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*
log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(
e*x + d) + (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^6 - (2*a^3*c^6*d^2 - 3*a^4*c^5*e^2
)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt((4*c^2*d
^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e
^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) - (a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5
- 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8
+ 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 +
162*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x + d) - (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*
d*e^6 - (2*a^3*c^6*d^2 - 3*a^4*c^5*e^2)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 +
81*a^2*e^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + a^3*c
^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3)))
+ (a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*
sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log(
-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x
+ d) + (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^6 + (2*a^3*c^6*d^2 - 3*a^4*c^5*e^2)*sq
rt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 -
15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 +
81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) - (a*c^2*x^2 - a^2*c)*sqrt((4*c^2*d^5 - 15
*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81
*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))*log(-(20*c^3*d^6*e^3 - 101*a*c^2*d^4*e^5 + 162
*a^2*c*d^2*e^7 - 81*a^3*e^9)*sqrt(e*x + d) - (5*a^2*c^3*d^3*e^4 - 9*a^3*c^2*d*e^
6 + (2*a^3*c^6*d^2 - 3*a^4*c^5*e^2)*sqrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a
^2*e^10)/(a^3*c^7)))*sqrt((4*c^2*d^5 - 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - a^3*c^3*s
qrt((25*c^2*d^4*e^6 - 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7)))/(a^3*c^3))) + 4*
(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d))/(a*c^2*x^2 - a^2*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)**(5/2)/(-c*x**2+a)**2,x)
[Out]
Timed out
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/(c*x^2 - a)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError