3.546 \(\int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^9} \, dx\)

Optimal. Leaf size=332 \[ -\frac{5 a^3 c^4 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{128 \left (a e^2+c d^2\right )^{11/2}}-\frac{5 a^2 c^3 \sqrt{a+c x^2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{128 (d+e x)^2 \left (a e^2+c d^2\right )^5}-\frac{5 a c^2 \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{192 (d+e x)^4 \left (a e^2+c d^2\right )^4}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 (d+e x)^7 \left (a e^2+c d^2\right )^2}-\frac{c \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{48 (d+e x)^6 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{7/2}}{8 (d+e x)^8 \left (a e^2+c d^2\right )} \]

[Out]

(-5*a^2*c^3*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(128*(c*d^2 + a*e^2
)^5*(d + e*x)^2) - (5*a*c^2*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(
192*(c*d^2 + a*e^2)^4*(d + e*x)^4) - (c*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x
^2)^(5/2))/(48*(c*d^2 + a*e^2)^3*(d + e*x)^6) - (e*(a + c*x^2)^(7/2))/(8*(c*d^2
+ a*e^2)*(d + e*x)^8) - (9*c*d*e*(a + c*x^2)^(7/2))/(56*(c*d^2 + a*e^2)^2*(d + e
*x)^7) - (5*a^3*c^4*(8*c*d^2 - a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]
*Sqrt[a + c*x^2])])/(128*(c*d^2 + a*e^2)^(11/2))

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Rubi [A]  time = 0.744564, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{5 a^3 c^4 \left (8 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{128 \left (a e^2+c d^2\right )^{11/2}}-\frac{5 a^2 c^3 \sqrt{a+c x^2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{128 (d+e x)^2 \left (a e^2+c d^2\right )^5}-\frac{5 a c^2 \left (a+c x^2\right )^{3/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{192 (d+e x)^4 \left (a e^2+c d^2\right )^4}-\frac{9 c d e \left (a+c x^2\right )^{7/2}}{56 (d+e x)^7 \left (a e^2+c d^2\right )^2}-\frac{c \left (a+c x^2\right )^{5/2} \left (8 c d^2-a e^2\right ) (a e-c d x)}{48 (d+e x)^6 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{7/2}}{8 (d+e x)^8 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(-5*a^2*c^3*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(128*(c*d^2 + a*e^2
)^5*(d + e*x)^2) - (5*a*c^2*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(
192*(c*d^2 + a*e^2)^4*(d + e*x)^4) - (c*(8*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x
^2)^(5/2))/(48*(c*d^2 + a*e^2)^3*(d + e*x)^6) - (e*(a + c*x^2)^(7/2))/(8*(c*d^2
+ a*e^2)*(d + e*x)^8) - (9*c*d*e*(a + c*x^2)^(7/2))/(56*(c*d^2 + a*e^2)^2*(d + e
*x)^7) - (5*a^3*c^4*(8*c*d^2 - a*e^2)*ArcTanh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]
*Sqrt[a + c*x^2])])/(128*(c*d^2 + a*e^2)^(11/2))

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Rubi in Sympy [A]  time = 73.5441, size = 316, normalized size = 0.95 \[ \frac{5 a^{3} c^{4} \left (\frac{a e^{2}}{8} - c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{16 \left (a e^{2} + c d^{2}\right )^{\frac{11}{2}}} + \frac{5 a^{2} c^{3} \sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right ) \left (\frac{a e^{2}}{8} - c d^{2}\right )}{32 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{5}} + \frac{5 a c^{2} \left (a + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - 2 c d x\right ) \left (a e^{2} - 8 c d^{2}\right )}{384 \left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )^{4}} - \frac{9 c d e \left (a + c x^{2}\right )^{\frac{7}{2}}}{56 \left (d + e x\right )^{7} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c \left (a + c x^{2}\right )^{\frac{5}{2}} \left (2 a e - 2 c d x\right ) \left (a e^{2} - 8 c d^{2}\right )}{96 \left (d + e x\right )^{6} \left (a e^{2} + c d^{2}\right )^{3}} - \frac{e \left (a + c x^{2}\right )^{\frac{7}{2}}}{8 \left (d + e x\right )^{8} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(5/2)/(e*x+d)**9,x)

[Out]

5*a**3*c**4*(a*e**2/8 - c*d**2)*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)**(11/2)) + 5*a**2*c**3*sqrt(a + c*x**2)*(2*a
*e - 2*c*d*x)*(a*e**2/8 - c*d**2)/(32*(d + e*x)**2*(a*e**2 + c*d**2)**5) + 5*a*c
**2*(a + c*x**2)**(3/2)*(2*a*e - 2*c*d*x)*(a*e**2 - 8*c*d**2)/(384*(d + e*x)**4*
(a*e**2 + c*d**2)**4) - 9*c*d*e*(a + c*x**2)**(7/2)/(56*(d + e*x)**7*(a*e**2 + c
*d**2)**2) + c*(a + c*x**2)**(5/2)*(2*a*e - 2*c*d*x)*(a*e**2 - 8*c*d**2)/(96*(d
+ e*x)**6*(a*e**2 + c*d**2)**3) - e*(a + c*x**2)**(7/2)/(8*(d + e*x)**8*(a*e**2
+ c*d**2))

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Mathematica [A]  time = 2.99636, size = 489, normalized size = 1.47 \[ \frac{\frac{105 a^3 c^4 \left (a e^2-8 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{11/2}}+\frac{105 a^3 c^4 \left (8 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{11/2}}-\frac{\sqrt{a+c x^2} \left (2 c^2 (d+e x)^4 \left (413 a^2 e^4+880 a c d^2 e^2+440 c^2 d^4\right ) \left (a e^2+c d^2\right )^3-2 c^3 d (d+e x)^5 \left (87 a^2 e^4+32 a c d^2 e^2+8 c^2 d^4\right ) \left (a e^2+c d^2\right )^2-c^3 (d+e x)^6 \left (-105 a^3 e^6+282 a^2 c d^2 e^4+88 a c^2 d^4 e^2+16 c^3 d^6\right ) \left (a e^2+c d^2\right )-c^4 d (d+e x)^7 \left (-663 a^3 e^6+370 a^2 c d^2 e^4+104 a c^2 d^4 e^2+16 c^3 d^6\right )-8 c^2 d (d+e x)^3 \left (307 a e^2+310 c d^2\right ) \left (a e^2+c d^2\right )^4-1584 c d (d+e x) \left (a e^2+c d^2\right )^6+8 c (d+e x)^2 \left (119 a e^2+362 c d^2\right ) \left (a e^2+c d^2\right )^5+336 \left (a e^2+c d^2\right )^7\right )}{(d+e x)^8 \left (a e^3+c d^2 e\right )^5}}{2688} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(5/2)/(d + e*x)^9,x]

[Out]

(-((Sqrt[a + c*x^2]*(336*(c*d^2 + a*e^2)^7 - 1584*c*d*(c*d^2 + a*e^2)^6*(d + e*x
) + 8*c*(c*d^2 + a*e^2)^5*(362*c*d^2 + 119*a*e^2)*(d + e*x)^2 - 8*c^2*d*(c*d^2 +
a*e^2)^4*(310*c*d^2 + 307*a*e^2)*(d + e*x)^3 + 2*c^2*(c*d^2 + a*e^2)^3*(440*c^2
*d^4 + 880*a*c*d^2*e^2 + 413*a^2*e^4)*(d + e*x)^4 - 2*c^3*d*(c*d^2 + a*e^2)^2*(8
*c^2*d^4 + 32*a*c*d^2*e^2 + 87*a^2*e^4)*(d + e*x)^5 - c^3*(c*d^2 + a*e^2)*(16*c^
3*d^6 + 88*a*c^2*d^4*e^2 + 282*a^2*c*d^2*e^4 - 105*a^3*e^6)*(d + e*x)^6 - c^4*d*
(16*c^3*d^6 + 104*a*c^2*d^4*e^2 + 370*a^2*c*d^2*e^4 - 663*a^3*e^6)*(d + e*x)^7))
/((c*d^2*e + a*e^3)^5*(d + e*x)^8)) + (105*a^3*c^4*(8*c*d^2 - a*e^2)*Log[d + e*x
])/(c*d^2 + a*e^2)^(11/2) + (105*a^3*c^4*(-8*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sq
rt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(11/2))/2688

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Maple [B]  time = 0.101, size = 9978, normalized size = 30.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)/(e*x+d)^9,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)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 31.7797, 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)^9,x, algorithm="fricas")

[Out]

[-1/5376*(2*(2616*a^3*c^4*d^8*e + 3865*a^4*c^3*d^6*e^3 + 3578*a^5*c^2*d^4*e^5 +
1720*a^6*c*d^2*e^7 + 336*a^7*e^9 - (16*c^7*d^7*e^2 + 104*a*c^6*d^5*e^4 + 370*a^2
*c^5*d^3*e^6 - 663*a^3*c^4*d*e^8)*x^7 - (128*c^7*d^8*e + 832*a*c^6*d^6*e^3 + 296
0*a^2*c^5*d^4*e^5 - 4464*a^3*c^4*d^2*e^7 - 105*a^4*c^3*e^9)*x^6 - (448*c^7*d^9 +
2904*a*c^6*d^7*e^2 + 10308*a^2*c^5*d^5*e^4 - 12449*a^3*c^4*d^3*e^6 - 456*a^4*c^
3*d*e^8)*x^5 - (1280*a*c^6*d^8*e + 11344*a^2*c^5*d^6*e^3 - 27128*a^3*c^4*d^4*e^5
- 4943*a^4*c^3*d^2*e^7 - 826*a^5*c^2*e^9)*x^4 - (1456*a*c^6*d^9 + 13250*a^2*c^5
*d^7*e^2 - 25441*a^3*c^4*d^5*e^4 - 5008*a^4*c^3*d^3*e^6 - 848*a^5*c^2*d*e^8)*x^3
- (4416*a^2*c^5*d^8*e - 23488*a^3*c^4*d^6*e^3 - 12351*a^4*c^3*d^4*e^5 - 5244*a^
5*c^2*d^2*e^7 - 952*a^6*c*e^9)*x^2 - (1848*a^2*c^5*d^9 - 7383*a^3*c^4*d^7*e^2 -
4040*a^4*c^3*d^5*e^4 - 1744*a^5*c^2*d^3*e^6 - 320*a^6*c*d*e^8)*x)*sqrt(c*d^2 + a
*e^2)*sqrt(c*x^2 + a) - 105*(8*a^3*c^5*d^10 - a^4*c^4*d^8*e^2 + (8*a^3*c^5*d^2*e
^8 - a^4*c^4*e^10)*x^8 + 8*(8*a^3*c^5*d^3*e^7 - a^4*c^4*d*e^9)*x^7 + 28*(8*a^3*c
^5*d^4*e^6 - a^4*c^4*d^2*e^8)*x^6 + 56*(8*a^3*c^5*d^5*e^5 - a^4*c^4*d^3*e^7)*x^5
+ 70*(8*a^3*c^5*d^6*e^4 - a^4*c^4*d^4*e^6)*x^4 + 56*(8*a^3*c^5*d^7*e^3 - a^4*c^
4*d^5*e^5)*x^3 + 28*(8*a^3*c^5*d^8*e^2 - a^4*c^4*d^6*e^4)*x^2 + 8*(8*a^3*c^5*d^9
*e - a^4*c^4*d^7*e^3)*x)*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)*sqrt(c*d^2 + a*e^2) + 2*(a*c*d^2*e + a^2*e^3 - (c^2*d^3 + a*c*d*e^
2)*x)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^5*d^18 + 5*a*c^4*d^16*e^2
+ 10*a^2*c^3*d^14*e^4 + 10*a^3*c^2*d^12*e^6 + 5*a^4*c*d^10*e^8 + a^5*d^8*e^10 +
(c^5*d^10*e^8 + 5*a*c^4*d^8*e^10 + 10*a^2*c^3*d^6*e^12 + 10*a^3*c^2*d^4*e^14 +
5*a^4*c*d^2*e^16 + a^5*e^18)*x^8 + 8*(c^5*d^11*e^7 + 5*a*c^4*d^9*e^9 + 10*a^2*c^
3*d^7*e^11 + 10*a^3*c^2*d^5*e^13 + 5*a^4*c*d^3*e^15 + a^5*d*e^17)*x^7 + 28*(c^5*
d^12*e^6 + 5*a*c^4*d^10*e^8 + 10*a^2*c^3*d^8*e^10 + 10*a^3*c^2*d^6*e^12 + 5*a^4*
c*d^4*e^14 + a^5*d^2*e^16)*x^6 + 56*(c^5*d^13*e^5 + 5*a*c^4*d^11*e^7 + 10*a^2*c^
3*d^9*e^9 + 10*a^3*c^2*d^7*e^11 + 5*a^4*c*d^5*e^13 + a^5*d^3*e^15)*x^5 + 70*(c^5
*d^14*e^4 + 5*a*c^4*d^12*e^6 + 10*a^2*c^3*d^10*e^8 + 10*a^3*c^2*d^8*e^10 + 5*a^4
*c*d^6*e^12 + a^5*d^4*e^14)*x^4 + 56*(c^5*d^15*e^3 + 5*a*c^4*d^13*e^5 + 10*a^2*c
^3*d^11*e^7 + 10*a^3*c^2*d^9*e^9 + 5*a^4*c*d^7*e^11 + a^5*d^5*e^13)*x^3 + 28*(c^
5*d^16*e^2 + 5*a*c^4*d^14*e^4 + 10*a^2*c^3*d^12*e^6 + 10*a^3*c^2*d^10*e^8 + 5*a^
4*c*d^8*e^10 + a^5*d^6*e^12)*x^2 + 8*(c^5*d^17*e + 5*a*c^4*d^15*e^3 + 10*a^2*c^3
*d^13*e^5 + 10*a^3*c^2*d^11*e^7 + 5*a^4*c*d^9*e^9 + a^5*d^7*e^11)*x)*sqrt(c*d^2
+ a*e^2)), -1/2688*((2616*a^3*c^4*d^8*e + 3865*a^4*c^3*d^6*e^3 + 3578*a^5*c^2*d^
4*e^5 + 1720*a^6*c*d^2*e^7 + 336*a^7*e^9 - (16*c^7*d^7*e^2 + 104*a*c^6*d^5*e^4 +
370*a^2*c^5*d^3*e^6 - 663*a^3*c^4*d*e^8)*x^7 - (128*c^7*d^8*e + 832*a*c^6*d^6*e
^3 + 2960*a^2*c^5*d^4*e^5 - 4464*a^3*c^4*d^2*e^7 - 105*a^4*c^3*e^9)*x^6 - (448*c
^7*d^9 + 2904*a*c^6*d^7*e^2 + 10308*a^2*c^5*d^5*e^4 - 12449*a^3*c^4*d^3*e^6 - 45
6*a^4*c^3*d*e^8)*x^5 - (1280*a*c^6*d^8*e + 11344*a^2*c^5*d^6*e^3 - 27128*a^3*c^4
*d^4*e^5 - 4943*a^4*c^3*d^2*e^7 - 826*a^5*c^2*e^9)*x^4 - (1456*a*c^6*d^9 + 13250
*a^2*c^5*d^7*e^2 - 25441*a^3*c^4*d^5*e^4 - 5008*a^4*c^3*d^3*e^6 - 848*a^5*c^2*d*
e^8)*x^3 - (4416*a^2*c^5*d^8*e - 23488*a^3*c^4*d^6*e^3 - 12351*a^4*c^3*d^4*e^5 -
5244*a^5*c^2*d^2*e^7 - 952*a^6*c*e^9)*x^2 - (1848*a^2*c^5*d^9 - 7383*a^3*c^4*d^
7*e^2 - 4040*a^4*c^3*d^5*e^4 - 1744*a^5*c^2*d^3*e^6 - 320*a^6*c*d*e^8)*x)*sqrt(-
c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 105*(8*a^3*c^5*d^10 - a^4*c^4*d^8*e^2 + (8*a^3*
c^5*d^2*e^8 - a^4*c^4*e^10)*x^8 + 8*(8*a^3*c^5*d^3*e^7 - a^4*c^4*d*e^9)*x^7 + 28
*(8*a^3*c^5*d^4*e^6 - a^4*c^4*d^2*e^8)*x^6 + 56*(8*a^3*c^5*d^5*e^5 - a^4*c^4*d^3
*e^7)*x^5 + 70*(8*a^3*c^5*d^6*e^4 - a^4*c^4*d^4*e^6)*x^4 + 56*(8*a^3*c^5*d^7*e^3
- a^4*c^4*d^5*e^5)*x^3 + 28*(8*a^3*c^5*d^8*e^2 - a^4*c^4*d^6*e^4)*x^2 + 8*(8*a^
3*c^5*d^9*e - a^4*c^4*d^7*e^3)*x)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*
d^2 + a*e^2)*sqrt(c*x^2 + a))))/((c^5*d^18 + 5*a*c^4*d^16*e^2 + 10*a^2*c^3*d^14*
e^4 + 10*a^3*c^2*d^12*e^6 + 5*a^4*c*d^10*e^8 + a^5*d^8*e^10 + (c^5*d^10*e^8 + 5*
a*c^4*d^8*e^10 + 10*a^2*c^3*d^6*e^12 + 10*a^3*c^2*d^4*e^14 + 5*a^4*c*d^2*e^16 +
a^5*e^18)*x^8 + 8*(c^5*d^11*e^7 + 5*a*c^4*d^9*e^9 + 10*a^2*c^3*d^7*e^11 + 10*a^3
*c^2*d^5*e^13 + 5*a^4*c*d^3*e^15 + a^5*d*e^17)*x^7 + 28*(c^5*d^12*e^6 + 5*a*c^4*
d^10*e^8 + 10*a^2*c^3*d^8*e^10 + 10*a^3*c^2*d^6*e^12 + 5*a^4*c*d^4*e^14 + a^5*d^
2*e^16)*x^6 + 56*(c^5*d^13*e^5 + 5*a*c^4*d^11*e^7 + 10*a^2*c^3*d^9*e^9 + 10*a^3*
c^2*d^7*e^11 + 5*a^4*c*d^5*e^13 + a^5*d^3*e^15)*x^5 + 70*(c^5*d^14*e^4 + 5*a*c^4
*d^12*e^6 + 10*a^2*c^3*d^10*e^8 + 10*a^3*c^2*d^8*e^10 + 5*a^4*c*d^6*e^12 + a^5*d
^4*e^14)*x^4 + 56*(c^5*d^15*e^3 + 5*a*c^4*d^13*e^5 + 10*a^2*c^3*d^11*e^7 + 10*a^
3*c^2*d^9*e^9 + 5*a^4*c*d^7*e^11 + a^5*d^5*e^13)*x^3 + 28*(c^5*d^16*e^2 + 5*a*c^
4*d^14*e^4 + 10*a^2*c^3*d^12*e^6 + 10*a^3*c^2*d^10*e^8 + 5*a^4*c*d^8*e^10 + a^5*
d^6*e^12)*x^2 + 8*(c^5*d^17*e + 5*a*c^4*d^15*e^3 + 10*a^2*c^3*d^13*e^5 + 10*a^3*
c^2*d^11*e^7 + 5*a^4*c*d^9*e^9 + a^5*d^7*e^11)*x)*sqrt(-c*d^2 - a*e^2))]

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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)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.406109, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(5/2)/(e*x + d)^9,x, algorithm="giac")

[Out]

Done