3.545 \(\int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=246 \[ -\frac{5 a^3 c^4 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{5 a^2 c^3 d \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac{5 a c^2 d \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac{c d \left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )} \]

[Out]

(-5*a^2*c^3*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^4*(d + e*x)^2)
- (5*a*c^2*d*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(24*(c*d^2 + a*e^2)^3*(d + e*x)^4)
- (c*d*(a*e - c*d*x)*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)^2*(d + e*x)^6) - (e*
(a + c*x^2)^(7/2))/(7*(c*d^2 + a*e^2)*(d + e*x)^7) - (5*a^3*c^4*d*ArcTanh[(a*e -
c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(9/2))

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Rubi [A]  time = 0.459973, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{5 a^3 c^4 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{5 a^2 c^3 d \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac{5 a c^2 d \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac{c d \left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{7/2}}{7 (d+e x)^7 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

(-5*a^2*c^3*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^4*(d + e*x)^2)
- (5*a*c^2*d*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(24*(c*d^2 + a*e^2)^3*(d + e*x)^4)
- (c*d*(a*e - c*d*x)*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)^2*(d + e*x)^6) - (e*
(a + c*x^2)^(7/2))/(7*(c*d^2 + a*e^2)*(d + e*x)^7) - (5*a^3*c^4*d*ArcTanh[(a*e -
c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(16*(c*d^2 + a*e^2)^(9/2))

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Rubi in Sympy [A]  time = 49.831, size = 240, normalized size = 0.98 \[ - \frac{5 a^{3} c^{4} d \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{16 \left (a e^{2} + c d^{2}\right )^{\frac{9}{2}}} - \frac{5 a^{2} c^{3} d \sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right )}{32 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{4}} - \frac{5 a c^{2} d \left (a + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - 2 c d x\right )}{48 \left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )^{3}} - \frac{c d \left (a + c x^{2}\right )^{\frac{5}{2}} \left (2 a e - 2 c d x\right )}{12 \left (d + e x\right )^{6} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{e \left (a + c x^{2}\right )^{\frac{7}{2}}}{7 \left (d + e x\right )^{7} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(5/2)/(e*x+d)**8,x)

[Out]

-5*a**3*c**4*d*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(16
*(a*e**2 + c*d**2)**(9/2)) - 5*a**2*c**3*d*sqrt(a + c*x**2)*(2*a*e - 2*c*d*x)/(3
2*(d + e*x)**2*(a*e**2 + c*d**2)**4) - 5*a*c**2*d*(a + c*x**2)**(3/2)*(2*a*e - 2
*c*d*x)/(48*(d + e*x)**4*(a*e**2 + c*d**2)**3) - c*d*(a + c*x**2)**(5/2)*(2*a*e
- 2*c*d*x)/(12*(d + e*x)**6*(a*e**2 + c*d**2)**2) - e*(a + c*x**2)**(7/2)/(7*(d
+ e*x)**7*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.968657, size = 403, normalized size = 1.64 \[ -\frac{5 a^3 c^4 d \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{16 \left (a e^2+c d^2\right )^{9/2}}+\frac{5 a^3 c^4 d \log (d+e x)}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{\sqrt{a+c x^2} \left (2 c^2 (d+e x)^4 \left (72 a^2 e^4+159 a c d^2 e^2+80 c^2 d^4\right ) \left (a e^2+c d^2\right )^2-c^3 d (d+e x)^5 \left (57 a^2 e^4+30 a c d^2 e^2+8 c^2 d^4\right ) \left (a e^2+c d^2\right )-c^3 (d+e x)^6 \left (-48 a^3 e^6+87 a^2 c d^2 e^4+38 a c^2 d^4 e^2+8 c^3 d^6\right )-2 c^2 d (d+e x)^3 \left (197 a e^2+200 c d^2\right ) \left (a e^2+c d^2\right )^3-232 c d (d+e x) \left (a e^2+c d^2\right )^5+8 c (d+e x)^2 \left (18 a e^2+55 c d^2\right ) \left (a e^2+c d^2\right )^4+48 \left (a e^2+c d^2\right )^6\right )}{336 e^5 (d+e x)^7 \left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(5/2)/(d + e*x)^8,x]

[Out]

-(Sqrt[a + c*x^2]*(48*(c*d^2 + a*e^2)^6 - 232*c*d*(c*d^2 + a*e^2)^5*(d + e*x) +
8*c*(c*d^2 + a*e^2)^4*(55*c*d^2 + 18*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e^2
)^3*(200*c*d^2 + 197*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2 + a*e^2)^2*(80*c^2*d^4 +
159*a*c*d^2*e^2 + 72*a^2*e^4)*(d + e*x)^4 - c^3*d*(c*d^2 + a*e^2)*(8*c^2*d^4 + 3
0*a*c*d^2*e^2 + 57*a^2*e^4)*(d + e*x)^5 - c^3*(8*c^3*d^6 + 38*a*c^2*d^4*e^2 + 87
*a^2*c*d^2*e^4 - 48*a^3*e^6)*(d + e*x)^6))/(336*e^5*(c*d^2 + a*e^2)^4*(d + e*x)^
7) + (5*a^3*c^4*d*Log[d + e*x])/(16*(c*d^2 + a*e^2)^(9/2)) - (5*a^3*c^4*d*Log[a*
e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(16*(c*d^2 + a*e^2)^(9/2))

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Maple [B]  time = 0.063, size = 7718, normalized size = 31.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)/(e*x+d)^8,x)

[Out]

result too large to display

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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)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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((c*x^2 + a)^(5/2)/(e*x + d)^8,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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((c*x**2+a)**(5/2)/(e*x+d)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.678763, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d)^8,x, algorithm="giac")

[Out]

sage0*x