3.533 \(\int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=269 \[ -\frac{a^2 c^3 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{a c^2 \sqrt{a+c x^2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 (d+e x)^5 \left (a e^2+c d^2\right )^2}-\frac{c \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

[Out]

-(a*c^2*(6*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^4*(
d + e*x)^2) - (c*(6*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(24*(c*d^2 +
a*e^2)^3*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)*(d + e*x)^6) -
(7*c*d*e*(a + c*x^2)^(5/2))/(30*(c*d^2 + a*e^2)^2*(d + e*x)^5) - (a^2*c^3*(6*c*
d^2 - a*e^2)*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 = 0.601535, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{a^2 c^3 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{9/2}}-\frac{a c^2 \sqrt{a+c x^2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^4}-\frac{7 c d e \left (a+c x^2\right )^{5/2}}{30 (d+e x)^5 \left (a e^2+c d^2\right )^2}-\frac{c \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right ) (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{5/2}}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

-(a*c^2*(6*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(16*(c*d^2 + a*e^2)^4*(
d + e*x)^2) - (c*(6*c*d^2 - a*e^2)*(a*e - c*d*x)*(a + c*x^2)^(3/2))/(24*(c*d^2 +
a*e^2)^3*(d + e*x)^4) - (e*(a + c*x^2)^(5/2))/(6*(c*d^2 + a*e^2)*(d + e*x)^6) -
(7*c*d*e*(a + c*x^2)^(5/2))/(30*(c*d^2 + a*e^2)^2*(d + e*x)^5) - (a^2*c^3*(6*c*
d^2 - a*e^2)*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 = 53.3072, size = 252, normalized size = 0.94 \[ \frac{3 a^{2} c^{3} \left (\frac{a e^{2}}{6} - c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{8 \left (a e^{2} + c d^{2}\right )^{\frac{9}{2}}} + \frac{a c^{2} \sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right ) \left (a e^{2} - 6 c d^{2}\right )}{32 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{4}} - \frac{7 c d e \left (a + c x^{2}\right )^{\frac{5}{2}}}{30 \left (d + e x\right )^{5} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c \left (a + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - 2 c d x\right ) \left (a e^{2} - 6 c d^{2}\right )}{48 \left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )^{3}} - \frac{e \left (a + c x^{2}\right )^{\frac{5}{2}}}{6 \left (d + e x\right )^{6} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(3/2)/(e*x+d)**7,x)

[Out]

3*a**2*c**3*(a*e**2/6 - c*d**2)*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**
2 + c*d**2)))/(8*(a*e**2 + c*d**2)**(9/2)) + a*c**2*sqrt(a + c*x**2)*(2*a*e - 2*
c*d*x)*(a*e**2 - 6*c*d**2)/(32*(d + e*x)**2*(a*e**2 + c*d**2)**4) - 7*c*d*e*(a +
c*x**2)**(5/2)/(30*(d + e*x)**5*(a*e**2 + c*d**2)**2) + c*(a + c*x**2)**(3/2)*(
2*a*e - 2*c*d*x)*(a*e**2 - 6*c*d**2)/(48*(d + e*x)**4*(a*e**2 + c*d**2)**3) - e*
(a + c*x**2)**(5/2)/(6*(d + e*x)**6*(a*e**2 + c*d**2))

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Mathematica [A]  time = 1.18251, size = 358, normalized size = 1.33 \[ \frac{1}{240} \left (\frac{15 a^2 c^3 \left (a e^2-6 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{9/2}}+\frac{15 a^2 c^3 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{9/2}}-\frac{\sqrt{a+c x^2} \left (-c^2 (d+e x)^4 \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right ) \left (a e^2+c d^2\right )-c^3 d (d+e x)^5 \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )-2 c^2 d (d+e x)^3 \left (9 a e^2+2 c d^2\right ) \left (a e^2+c d^2\right )^2-104 c d (d+e x) \left (a e^2+c d^2\right )^4+2 c (d+e x)^2 \left (35 a e^2+38 c d^2\right ) \left (a e^2+c d^2\right )^3+40 \left (a e^2+c d^2\right )^5\right )}{e^3 (d+e x)^6 \left (a e^2+c d^2\right )^4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(3/2)/(d + e*x)^7,x]

[Out]

(-((Sqrt[a + c*x^2]*(40*(c*d^2 + a*e^2)^5 - 104*c*d*(c*d^2 + a*e^2)^4*(d + e*x)
+ 2*c*(c*d^2 + a*e^2)^3*(38*c*d^2 + 35*a*e^2)*(d + e*x)^2 - 2*c^2*d*(c*d^2 + a*e
^2)^2*(2*c*d^2 + 9*a*e^2)*(d + e*x)^3 - c^2*(c*d^2 + a*e^2)*(4*c^2*d^4 + 24*a*c*
d^2*e^2 - 15*a^2*e^4)*(d + e*x)^4 - c^3*d*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*a^2*e
^4)*(d + e*x)^5))/(e^3*(c*d^2 + a*e^2)^4*(d + e*x)^6)) + (15*a^2*c^3*(6*c*d^2 -
a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(9/2) + (15*a^2*c^3*(-6*c*d^2 + a*e^2)*Log[
a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9/2))/240

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Maple [B]  time = 0.041, size = 5087, normalized size = 18.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(3/2)/(e*x+d)^7,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)^(3/2)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 8.14284, 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)^(3/2)/(e*x + d)^7,x, algorithm="fricas")

[Out]

[-1/480*(2*(246*a^2*c^3*d^6*e + 267*a^3*c^2*d^4*e^3 + 166*a^4*c*d^2*e^5 + 40*a^5
*e^7 - (4*c^5*d^5*e^2 + 28*a*c^4*d^3*e^4 - 81*a^2*c^3*d*e^6)*x^5 - 3*(8*c^5*d^6*
e + 56*a*c^4*d^4*e^3 - 132*a^2*c^3*d^2*e^5 - 5*a^3*c^2*e^7)*x^4 - 2*(30*c^5*d^7
+ 209*a*c^4*d^5*e^2 - 367*a^2*c^3*d^3*e^4 - 21*a^3*c^2*d*e^6)*x^3 - 2*(114*a*c^4
*d^6*e - 537*a^2*c^3*d^4*e^3 - 161*a^3*c^2*d^2*e^5 - 35*a^4*c*e^7)*x^2 - 3*(50*a
*c^4*d^7 - 167*a^2*c^3*d^5*e^2 - 54*a^3*c^2*d^3*e^4 - 12*a^4*c*d*e^6)*x)*sqrt(c*
d^2 + a*e^2)*sqrt(c*x^2 + a) + 15*(6*a^2*c^4*d^8 - a^3*c^3*d^6*e^2 + (6*a^2*c^4*
d^2*e^6 - a^3*c^3*e^8)*x^6 + 6*(6*a^2*c^4*d^3*e^5 - a^3*c^3*d*e^7)*x^5 + 15*(6*a
^2*c^4*d^4*e^4 - a^3*c^3*d^2*e^6)*x^4 + 20*(6*a^2*c^4*d^5*e^3 - a^3*c^3*d^3*e^5)
*x^3 + 15*(6*a^2*c^4*d^6*e^2 - a^3*c^3*d^4*e^4)*x^2 + 6*(6*a^2*c^4*d^7*e - a^3*c
^3*d^5*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^4*d^14 + 4*a*c^3*d^12*e^2 + 6*a^2*c
^2*d^10*e^4 + 4*a^3*c*d^8*e^6 + a^4*d^6*e^8 + (c^4*d^8*e^6 + 4*a*c^3*d^6*e^8 + 6
*a^2*c^2*d^4*e^10 + 4*a^3*c*d^2*e^12 + a^4*e^14)*x^6 + 6*(c^4*d^9*e^5 + 4*a*c^3*
d^7*e^7 + 6*a^2*c^2*d^5*e^9 + 4*a^3*c*d^3*e^11 + a^4*d*e^13)*x^5 + 15*(c^4*d^10*
e^4 + 4*a*c^3*d^8*e^6 + 6*a^2*c^2*d^6*e^8 + 4*a^3*c*d^4*e^10 + a^4*d^2*e^12)*x^4
+ 20*(c^4*d^11*e^3 + 4*a*c^3*d^9*e^5 + 6*a^2*c^2*d^7*e^7 + 4*a^3*c*d^5*e^9 + a^
4*d^3*e^11)*x^3 + 15*(c^4*d^12*e^2 + 4*a*c^3*d^10*e^4 + 6*a^2*c^2*d^8*e^6 + 4*a^
3*c*d^6*e^8 + a^4*d^4*e^10)*x^2 + 6*(c^4*d^13*e + 4*a*c^3*d^11*e^3 + 6*a^2*c^2*d
^9*e^5 + 4*a^3*c*d^7*e^7 + a^4*d^5*e^9)*x)*sqrt(c*d^2 + a*e^2)), -1/240*((246*a^
2*c^3*d^6*e + 267*a^3*c^2*d^4*e^3 + 166*a^4*c*d^2*e^5 + 40*a^5*e^7 - (4*c^5*d^5*
e^2 + 28*a*c^4*d^3*e^4 - 81*a^2*c^3*d*e^6)*x^5 - 3*(8*c^5*d^6*e + 56*a*c^4*d^4*e
^3 - 132*a^2*c^3*d^2*e^5 - 5*a^3*c^2*e^7)*x^4 - 2*(30*c^5*d^7 + 209*a*c^4*d^5*e^
2 - 367*a^2*c^3*d^3*e^4 - 21*a^3*c^2*d*e^6)*x^3 - 2*(114*a*c^4*d^6*e - 537*a^2*c
^3*d^4*e^3 - 161*a^3*c^2*d^2*e^5 - 35*a^4*c*e^7)*x^2 - 3*(50*a*c^4*d^7 - 167*a^2
*c^3*d^5*e^2 - 54*a^3*c^2*d^3*e^4 - 12*a^4*c*d*e^6)*x)*sqrt(-c*d^2 - a*e^2)*sqrt
(c*x^2 + a) - 15*(6*a^2*c^4*d^8 - a^3*c^3*d^6*e^2 + (6*a^2*c^4*d^2*e^6 - a^3*c^3
*e^8)*x^6 + 6*(6*a^2*c^4*d^3*e^5 - a^3*c^3*d*e^7)*x^5 + 15*(6*a^2*c^4*d^4*e^4 -
a^3*c^3*d^2*e^6)*x^4 + 20*(6*a^2*c^4*d^5*e^3 - a^3*c^3*d^3*e^5)*x^3 + 15*(6*a^2*
c^4*d^6*e^2 - a^3*c^3*d^4*e^4)*x^2 + 6*(6*a^2*c^4*d^7*e - a^3*c^3*d^5*e^3)*x)*ar
ctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((c^
4*d^14 + 4*a*c^3*d^12*e^2 + 6*a^2*c^2*d^10*e^4 + 4*a^3*c*d^8*e^6 + a^4*d^6*e^8 +
(c^4*d^8*e^6 + 4*a*c^3*d^6*e^8 + 6*a^2*c^2*d^4*e^10 + 4*a^3*c*d^2*e^12 + a^4*e^
14)*x^6 + 6*(c^4*d^9*e^5 + 4*a*c^3*d^7*e^7 + 6*a^2*c^2*d^5*e^9 + 4*a^3*c*d^3*e^1
1 + a^4*d*e^13)*x^5 + 15*(c^4*d^10*e^4 + 4*a*c^3*d^8*e^6 + 6*a^2*c^2*d^6*e^8 + 4
*a^3*c*d^4*e^10 + a^4*d^2*e^12)*x^4 + 20*(c^4*d^11*e^3 + 4*a*c^3*d^9*e^5 + 6*a^2
*c^2*d^7*e^7 + 4*a^3*c*d^5*e^9 + a^4*d^3*e^11)*x^3 + 15*(c^4*d^12*e^2 + 4*a*c^3*
d^10*e^4 + 6*a^2*c^2*d^8*e^6 + 4*a^3*c*d^6*e^8 + a^4*d^4*e^10)*x^2 + 6*(c^4*d^13
*e + 4*a*c^3*d^11*e^3 + 6*a^2*c^2*d^9*e^5 + 4*a^3*c*d^7*e^7 + a^4*d^5*e^9)*x)*sq
rt(-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)**(3/2)/(e*x+d)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.298582, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^7,x, algorithm="giac")

[Out]

Done