3.539 \(\int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^2} \, dx\)
Optimal. Leaf size=219 \[ \frac{5 \sqrt{c} \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 e^6}+\frac{5 c d \left (a e^2+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^6}-\frac{5 c \sqrt{a+c x^2} \left (8 d \left (a e^2+c d^2\right )-e x \left (3 a e^2+4 c d^2\right )\right )}{8 e^5}-\frac{5 c \left (a+c x^2\right )^{3/2} (4 d-3 e x)}{12 e^3}-\frac{\left (a+c x^2\right )^{5/2}}{e (d+e x)} \]
[Out]
(-5*c*(8*d*(c*d^2 + a*e^2) - e*(4*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5) -
(5*c*(4*d - 3*e*x)*(a + c*x^2)^(3/2))/(12*e^3) - (a + c*x^2)^(5/2)/(e*(d + e*x)
) + (5*Sqrt[c]*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt
[a + c*x^2]])/(8*e^6) + (5*c*d*(c*d^2 + a*e^2)^(3/2)*ArcTanh[(a*e - c*d*x)/(Sqrt
[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/e^6
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Rubi [A] time = 0.665452, antiderivative size = 219, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316
\[ \frac{5 \sqrt{c} \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 e^6}+\frac{5 c d \left (a e^2+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^6}-\frac{5 c \sqrt{a+c x^2} \left (8 d \left (a e^2+c d^2\right )-e x \left (3 a e^2+4 c d^2\right )\right )}{8 e^5}-\frac{5 c \left (a+c x^2\right )^{3/2} (4 d-3 e x)}{12 e^3}-\frac{\left (a+c x^2\right )^{5/2}}{e (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^(5/2)/(d + e*x)^2,x]
[Out]
(-5*c*(8*d*(c*d^2 + a*e^2) - e*(4*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5) -
(5*c*(4*d - 3*e*x)*(a + c*x^2)^(3/2))/(12*e^3) - (a + c*x^2)^(5/2)/(e*(d + e*x)
) + (5*Sqrt[c]*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*a^2*e^4)*ArcTanh[(Sqrt[c]*x)/Sqrt
[a + c*x^2]])/(8*e^6) + (5*c*d*(c*d^2 + a*e^2)^(3/2)*ArcTanh[(a*e - c*d*x)/(Sqrt
[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/e^6
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Rubi in Sympy [A] time = 79.3309, size = 207, normalized size = 0.95 \[ \frac{5 \sqrt{c} \left (3 a^{2} e^{4} + 12 a c d^{2} e^{2} + 8 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 e^{6}} + \frac{5 c d \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{6}} - \frac{5 c \left (a + c x^{2}\right )^{\frac{3}{2}} \left (4 d - 3 e x\right )}{12 e^{3}} - \frac{5 c \sqrt{a + c x^{2}} \left (8 d \left (a e^{2} + c d^{2}\right ) - e x \left (3 a e^{2} + 4 c d^{2}\right )\right )}{8 e^{5}} - \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{e \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**(5/2)/(e*x+d)**2,x)
[Out]
5*sqrt(c)*(3*a**2*e**4 + 12*a*c*d**2*e**2 + 8*c**2*d**4)*atanh(sqrt(c)*x/sqrt(a
+ c*x**2))/(8*e**6) + 5*c*d*(a*e**2 + c*d**2)**(3/2)*atanh((a*e - c*d*x)/(sqrt(a
+ c*x**2)*sqrt(a*e**2 + c*d**2)))/e**6 - 5*c*(a + c*x**2)**(3/2)*(4*d - 3*e*x)/
(12*e**3) - 5*c*sqrt(a + c*x**2)*(8*d*(a*e**2 + c*d**2) - e*x*(3*a*e**2 + 4*c*d*
*2))/(8*e**5) - (a + c*x**2)**(5/2)/(e*(d + e*x))
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Mathematica [A] time = 0.36501, size = 239, normalized size = 1.09 \[ \frac{15 \sqrt{c} \left (3 a^2 e^4+12 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+e \sqrt{a+c x^2} \left (9 c e x \left (3 a e^2+4 c d^2\right )-\frac{24 \left (a e^2+c d^2\right )^2}{d+e x}-16 c d \left (7 a e^2+6 c d^2\right )-16 c^2 d e^2 x^2+6 c^2 e^3 x^3\right )+120 c d \left (a e^2+c d^2\right )^{3/2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )-120 c d \left (a e^2+c d^2\right )^{3/2} \log (d+e x)}{24 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^(5/2)/(d + e*x)^2,x]
[Out]
(e*Sqrt[a + c*x^2]*(-16*c*d*(6*c*d^2 + 7*a*e^2) + 9*c*e*(4*c*d^2 + 3*a*e^2)*x -
16*c^2*d*e^2*x^2 + 6*c^2*e^3*x^3 - (24*(c*d^2 + a*e^2)^2)/(d + e*x)) - 120*c*d*(
c*d^2 + a*e^2)^(3/2)*Log[d + e*x] + 15*Sqrt[c]*(8*c^2*d^4 + 12*a*c*d^2*e^2 + 3*a
^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] + 120*c*d*(c*d^2 + a*e^2)^(3/2)*Log[a
*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(24*e^6)
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Maple [B] time = 0.017, size = 1796, normalized size = 8.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^(5/2)/(e*x+d)^2,x)
[Out]
-1/(a*e^2+c*d^2)/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(7/2)-1
/e*c*d/(a*e^2+c*d^2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)+5/4/e
^2*c^2*d^2/(a*e^2+c*d^2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x
+35/8/e^2*c^2*d^2/(a*e^2+c*d^2)*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2
)^(1/2)*x+75/8/e^2*c^(3/2)*d^2/(a*e^2+c*d^2)*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d
/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))*a^2-5/3/e*c*d/(a*e^2+c*d^2)*(c
*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*a-5/3/e^3*c^2*d^3/(a*e^2+c*d
^2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)+5/2/e^4*c^3*d^4/(a*e^2
+c*d^2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+25/2/e^4*c^(5/2)
*d^4/(a*e^2+c*d^2)*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a
*e^2+c*d^2)/e^2)^(1/2))*a-5/e*c*d/(a*e^2+c*d^2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*
e^2+c*d^2)/e^2)^(1/2)*a^2-10/e^3*c^2*d^3/(a*e^2+c*d^2)*(c*(d/e+x)^2-2*c*d/e*(d/e
+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a-5/e^5*c^3*d^5/(a*e^2+c*d^2)*(c*(d/e+x)^2-2*c*d/e*
(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+5/e^6*c^(7/2)*d^6/(a*e^2+c*d^2)*ln((-c*d/e+c*(d
/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))+5/e*c*d/(a
*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*
((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))
/(d/e+x))*a^3+15/e^3*c^2*d^3/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^
2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d
/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a^2+15/e^5*c^3*d^5/(a*e^2+c*d^2)/((a*e^
2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2
)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))*a+5/e^7*
c^4*d^7/(a*e^2+c*d^2)/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*
(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e
^2)^(1/2))/(d/e+x))+1/(a*e^2+c*d^2)*c*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)
/e^2)^(5/2)*x+5/4/(a*e^2+c*d^2)*c*a*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e
^2)^(3/2)*x+15/8/(a*e^2+c*d^2)*c*a^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/
e^2)^(1/2)*x+15/8/(a*e^2+c*d^2)*c^(1/2)*a^3*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/
e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)/(e*x + d)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 13.9239, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)/(e*x + d)^2,x, algorithm="fricas")
[Out]
[1/48*(15*(8*c^2*d^5 + 12*a*c*d^3*e^2 + 3*a^2*d*e^4 + (8*c^2*d^4*e + 12*a*c*d^2*
e^3 + 3*a^2*e^5)*x)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 12
0*(c^2*d^4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d^2 + a*e^2)*log((2
*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 + 2*sqrt(c*d^2 + a*
e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(6*c^2*e^5*x^
4 - 10*c^2*d*e^4*x^3 - 120*c^2*d^4*e - 160*a*c*d^2*e^3 - 24*a^2*e^5 + (20*c^2*d^
2*e^3 + 27*a*c*e^5)*x^2 - 5*(12*c^2*d^3*e^2 + 17*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/
(e^7*x + d*e^6), -1/48*(240*(c^2*d^4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*
sqrt(-c*d^2 - a*e^2)*arctan((c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))
) - 15*(8*c^2*d^5 + 12*a*c*d^3*e^2 + 3*a^2*d*e^4 + (8*c^2*d^4*e + 12*a*c*d^2*e^3
+ 3*a^2*e^5)*x)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(6*
c^2*e^5*x^4 - 10*c^2*d*e^4*x^3 - 120*c^2*d^4*e - 160*a*c*d^2*e^3 - 24*a^2*e^5 +
(20*c^2*d^2*e^3 + 27*a*c*e^5)*x^2 - 5*(12*c^2*d^3*e^2 + 17*a*c*d*e^4)*x)*sqrt(c*
x^2 + a))/(e^7*x + d*e^6), 1/24*(15*(8*c^2*d^5 + 12*a*c*d^3*e^2 + 3*a^2*d*e^4 +
(8*c^2*d^4*e + 12*a*c*d^2*e^3 + 3*a^2*e^5)*x)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 +
a)*sqrt(-c))) + 60*(c^2*d^4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*sqrt(c*d^
2 + a*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*a^2*e^2 - (2*c^2*d^2 + a*c*e^2)*x^2 +
2*sqrt(c*d^2 + a*e^2)*(c*d*x - a*e)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2))
+ (6*c^2*e^5*x^4 - 10*c^2*d*e^4*x^3 - 120*c^2*d^4*e - 160*a*c*d^2*e^3 - 24*a^2*e
^5 + (20*c^2*d^2*e^3 + 27*a*c*e^5)*x^2 - 5*(12*c^2*d^3*e^2 + 17*a*c*d*e^4)*x)*sq
rt(c*x^2 + a))/(e^7*x + d*e^6), 1/24*(15*(8*c^2*d^5 + 12*a*c*d^3*e^2 + 3*a^2*d*e
^4 + (8*c^2*d^4*e + 12*a*c*d^2*e^3 + 3*a^2*e^5)*x)*sqrt(-c)*arctan(c*x/(sqrt(c*x
^2 + a)*sqrt(-c))) - 120*(c^2*d^4 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*sqr
t(-c*d^2 - a*e^2)*arctan((c*d*x - a*e)/(sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a))) +
(6*c^2*e^5*x^4 - 10*c^2*d*e^4*x^3 - 120*c^2*d^4*e - 160*a*c*d^2*e^3 - 24*a^2*e^
5 + (20*c^2*d^2*e^3 + 27*a*c*e^5)*x^2 - 5*(12*c^2*d^3*e^2 + 17*a*c*d*e^4)*x)*sqr
t(c*x^2 + a))/(e^7*x + d*e^6)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**(5/2)/(e*x+d)**2,x)
[Out]
Integral((a + c*x**2)**(5/2)/(d + e*x)**2, x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^(5/2)/(e*x + d)^2,x, algorithm="giac")
[Out]
undef