3.538 \(\int \frac{\left (a+c x^2\right )^{5/2}}{d+e x} \, dx\)

Optimal. Leaf size=226 \[ -\frac{\sqrt{c} d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 e^6}-\frac{\left (a e^2+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^6}+\frac{\sqrt{a+c x^2} \left (8 \left (a e^2+c d^2\right )^2-c d e x \left (7 a e^2+4 c d^2\right )\right )}{8 e^5}+\frac{\left (a+c x^2\right )^{3/2} \left (4 \left (a e^2+c d^2\right )-3 c d e x\right )}{12 e^3}+\frac{\left (a+c x^2\right )^{5/2}}{5 e} \]

[Out]

((8*(c*d^2 + a*e^2)^2 - c*d*e*(4*c*d^2 + 7*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5) +
((4*(c*d^2 + a*e^2) - 3*c*d*e*x)*(a + c*x^2)^(3/2))/(12*e^3) + (a + c*x^2)^(5/2)
/(5*e) - (Sqrt[c]*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*ArcTanh[(Sqrt[c]*x
)/Sqrt[a + c*x^2]])/(8*e^6) - ((c*d^2 + a*e^2)^(5/2)*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 = 0.712635, antiderivative size = 226, 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 \[ -\frac{\sqrt{c} d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 e^6}-\frac{\left (a e^2+c d^2\right )^{5/2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^6}+\frac{\sqrt{a+c x^2} \left (8 \left (a e^2+c d^2\right )^2-c d e x \left (7 a e^2+4 c d^2\right )\right )}{8 e^5}+\frac{\left (a+c x^2\right )^{3/2} \left (4 \left (a e^2+c d^2\right )-3 c d e x\right )}{12 e^3}+\frac{\left (a+c x^2\right )^{5/2}}{5 e} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(5/2)/(d + e*x),x]

[Out]

((8*(c*d^2 + a*e^2)^2 - c*d*e*(4*c*d^2 + 7*a*e^2)*x)*Sqrt[a + c*x^2])/(8*e^5) +
((4*(c*d^2 + a*e^2) - 3*c*d*e*x)*(a + c*x^2)^(3/2))/(12*e^3) + (a + c*x^2)^(5/2)
/(5*e) - (Sqrt[c]*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 15*a^2*e^4)*ArcTanh[(Sqrt[c]*x
)/Sqrt[a + c*x^2]])/(8*e^6) - ((c*d^2 + a*e^2)^(5/2)*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 = 88.02, size = 209, normalized size = 0.92 \[ - \frac{\sqrt{c} d \left (15 a^{2} e^{4} + 20 a c d^{2} e^{2} + 8 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 e^{6}} + \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{5 e} + \frac{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (4 a e^{2} + 4 c d^{2} - 3 c d e x\right )}{12 e^{3}} - \frac{\sqrt{a + c x^{2}} \left (c d e x \left (7 a e^{2} + 4 c d^{2}\right ) - 8 \left (a e^{2} + c d^{2}\right )^{2}\right )}{8 e^{5}} - \frac{\left (a e^{2} + c d^{2}\right )^{\frac{5}{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(5/2)/(e*x+d),x)

[Out]

-sqrt(c)*d*(15*a**2*e**4 + 20*a*c*d**2*e**2 + 8*c**2*d**4)*atanh(sqrt(c)*x/sqrt(
a + c*x**2))/(8*e**6) + (a + c*x**2)**(5/2)/(5*e) + (a + c*x**2)**(3/2)*(4*a*e**
2 + 4*c*d**2 - 3*c*d*e*x)/(12*e**3) - sqrt(a + c*x**2)*(c*d*e*x*(7*a*e**2 + 4*c*
d**2) - 8*(a*e**2 + c*d**2)**2)/(8*e**5) - (a*e**2 + c*d**2)**(5/2)*atanh((a*e -
c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/e**6

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Mathematica [A]  time = 0.248258, size = 235, normalized size = 1.04 \[ \frac{-15 \sqrt{c} d \left (15 a^2 e^4+20 a c d^2 e^2+8 c^2 d^4\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+e \sqrt{a+c x^2} \left (184 a^2 e^4+a c e^2 \left (280 d^2-135 d e x+88 e^2 x^2\right )+2 c^2 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )-120 \left (a e^2+c d^2\right )^{5/2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+120 \left (a e^2+c d^2\right )^{5/2} \log (d+e x)}{120 e^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(5/2)/(d + e*x),x]

[Out]

(e*Sqrt[a + c*x^2]*(184*a^2*e^4 + a*c*e^2*(280*d^2 - 135*d*e*x + 88*e^2*x^2) + 2
*c^2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 120*(
c*d^2 + a*e^2)^(5/2)*Log[d + e*x] - 15*Sqrt[c]*d*(8*c^2*d^4 + 20*a*c*d^2*e^2 + 1
5*a^2*e^4)*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]] - 120*(c*d^2 + a*e^2)^(5/2)*Log[a*
e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(120*e^6)

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Maple [B]  time = 0.012, size = 1225, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)/(e*x+d),x)

[Out]

1/5/e*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-1/4/e^2*c*d*(c*(d/e+
x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x-7/8/e^2*c*d*a*(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/e^2*c^(1/2)*d*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+1/3/e*(c*(d/e+
x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*a+1/3/e^3*(c*(d/e+x)^2-2*c*d/e*(d/
e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*c*d^2-1/2/e^4*c^2*d^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)
+(a*e^2+c*d^2)/e^2)^(1/2)*x-5/2/e^4*c^(3/2)*d^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))*a+1/e*(c*(d/e+x)^2-2*c*d/e*
(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a^2+2/e^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c
*d^2)/e^2)^(1/2)*a*c*d^2+1/e^5*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(
1/2)*c^2*d^4-1/e^6*c^(5/2)*d^5*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))-1/e/((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^3-3/e^3/((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*c*d^2-3/e^5/((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*c^2*d^4-1/e^7/((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))*c^3*d^
6

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 56.9594, 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),x, algorithm="fricas")

[Out]

[1/240*(15*(8*c^2*d^5 + 20*a*c*d^3*e^2 + 15*a^2*d*e^4)*sqrt(c)*log(-2*c*x^2 + 2*
sqrt(c*x^2 + a)*sqrt(c)*x - a) + 120*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*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*(24*c^2*e^5*x^4 - 30*c^2*d*e^4*x^3 + 120*c^2*d^4*e + 280*a*c*d^2*e^3 + 184
*a^2*e^5 + 8*(5*c^2*d^2*e^3 + 11*a*c*e^5)*x^2 - 15*(4*c^2*d^3*e^2 + 9*a*c*d*e^4)
*x)*sqrt(c*x^2 + a))/e^6, -1/120*(15*(8*c^2*d^5 + 20*a*c*d^3*e^2 + 15*a^2*d*e^4)
*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) - 60*(c^2*d^4 + 2*a*c*d^2*e^2 +
a^2*e^4)*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)) - (24*c^2*e^5*x^4 - 30*c^2*d*e^4*x^3 + 120*c^2*d^4*e + 280*a*
c*d^2*e^3 + 184*a^2*e^5 + 8*(5*c^2*d^2*e^3 + 11*a*c*e^5)*x^2 - 15*(4*c^2*d^3*e^2
+ 9*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/e^6, 1/240*(240*(c^2*d^4 + 2*a*c*d^2*e^2 + a
^2*e^4)*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 + 20*a*c*d^3*e^2 + 15*a^2*d*e^4)*sqrt(c)*log(-2*c*x^2
+ 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(24*c^2*e^5*x^4 - 30*c^2*d*e^4*x^3 + 120*
c^2*d^4*e + 280*a*c*d^2*e^3 + 184*a^2*e^5 + 8*(5*c^2*d^2*e^3 + 11*a*c*e^5)*x^2 -
15*(4*c^2*d^3*e^2 + 9*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/e^6, -1/120*(15*(8*c^2*d^5
+ 20*a*c*d^3*e^2 + 15*a^2*d*e^4)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))
) - 120*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*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))) - (24*c^2*e^5*x^4 - 30*c^2*d*e^4*x
^3 + 120*c^2*d^4*e + 280*a*c*d^2*e^3 + 184*a^2*e^5 + 8*(5*c^2*d^2*e^3 + 11*a*c*e
^5)*x^2 - 15*(4*c^2*d^3*e^2 + 9*a*c*d*e^4)*x)*sqrt(c*x^2 + a))/e^6]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(5/2)/(e*x+d),x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x), x)

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GIAC/XCAS [A]  time = 0.224905, size = 381, normalized size = 1.69 \[ \frac{1}{8} \,{\left (8 \, c^{\frac{5}{2}} d^{5} + 20 \, a c^{\frac{3}{2}} d^{3} e^{2} + 15 \, a^{2} \sqrt{c} d e^{4}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right ) + \frac{2 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{\left (-6\right )}}{\sqrt{-c d^{2} - a e^{2}}} + \frac{1}{120} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (4 \, c^{2} x e^{\left (-1\right )} - 5 \, c^{2} d e^{\left (-2\right )}\right )} x + \frac{4 \,{\left (5 \, c^{5} d^{2} e^{18} + 11 \, a c^{4} e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x - \frac{15 \,{\left (4 \, c^{5} d^{3} e^{17} + 9 \, a c^{4} d e^{19}\right )} e^{\left (-21\right )}}{c^{3}}\right )} x + \frac{8 \,{\left (15 \, c^{5} d^{4} e^{16} + 35 \, a c^{4} d^{2} e^{18} + 23 \, a^{2} c^{3} e^{20}\right )} e^{\left (-21\right )}}{c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d),x, algorithm="giac")

[Out]

1/8*(8*c^(5/2)*d^5 + 20*a*c^(3/2)*d^3*e^2 + 15*a^2*sqrt(c)*d*e^4)*e^(-6)*ln(abs(
-sqrt(c)*x + sqrt(c*x^2 + a))) + 2*(c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4
+ a^3*e^6)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + a))*e + sqrt(c)*d)/sqrt(-c*d^2 - a
*e^2))*e^(-6)/sqrt(-c*d^2 - a*e^2) + 1/120*sqrt(c*x^2 + a)*((2*(3*(4*c^2*x*e^(-1
) - 5*c^2*d*e^(-2))*x + 4*(5*c^5*d^2*e^18 + 11*a*c^4*e^20)*e^(-21)/c^3)*x - 15*(
4*c^5*d^3*e^17 + 9*a*c^4*d*e^19)*e^(-21)/c^3)*x + 8*(15*c^5*d^4*e^16 + 35*a*c^4*
d^2*e^18 + 23*a^2*c^3*e^20)*e^(-21)/c^3)