3.544 \(\int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^7} \, dx\)

Optimal. Leaf size=203 \[ -\frac{5 a^3 c^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{7/2}}-\frac{5 a^2 c^2 \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{5 a c \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.325709, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{5 a^3 c^3 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{16 \left (a e^2+c d^2\right )^{7/2}}-\frac{5 a^2 c^2 \sqrt{a+c x^2} (a e-c d x)}{16 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{5 a c \left (a+c x^2\right )^{3/2} (a e-c d x)}{24 (d+e x)^4 \left (a e^2+c d^2\right )^2}-\frac{\left (a+c x^2\right )^{5/2} (a e-c d x)}{6 (d+e x)^6 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

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

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Rubi in Sympy [A]  time = 35.5337, size = 197, normalized size = 0.97 \[ - \frac{5 a^{3} c^{3} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{16 \left (a e^{2} + c d^{2}\right )^{\frac{7}{2}}} - \frac{5 a^{2} c^{2} \sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right )}{32 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{3}} - \frac{5 a c \left (a + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - 2 c d x\right )}{48 \left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{\left (a + c x^{2}\right )^{\frac{5}{2}} \left (2 a e - 2 c d x\right )}{12 \left (d + e x\right )^{6} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(5/2)/(e*x+d)**7,x)

[Out]

-5*a**3*c**3*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)**(7/2)) - 5*a**2*c**2*sqrt(a + c*x**2)*(2*a*e - 2*c*d*x)/(32*(d
+ e*x)**2*(a*e**2 + c*d**2)**3) - 5*a*c*(a + c*x**2)**(3/2)*(2*a*e - 2*c*d*x)/(
48*(d + e*x)**4*(a*e**2 + c*d**2)**2) - (a + c*x**2)**(5/2)*(2*a*e - 2*c*d*x)/(1
2*(d + e*x)**6*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.980628, size = 305, normalized size = 1.5 \[ \frac{1}{48} \left (-\frac{15 a^3 c^3 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac{15 a^3 c^3 \log (d+e x)}{\left (a e^2+c d^2\right )^{7/2}}+\frac{\sqrt{a+c x^2} \left (-8 a^5 e^5-2 a^4 c e^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )-a^3 c^2 e \left (33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right )+a^2 c^3 d x \left (33 d^4+54 d^3 e x+122 d^2 e^2 x^2+54 d e^3 x^3+33 e^4 x^4\right )+2 a c^4 d^3 x^3 \left (13 d^2+6 d e x+13 e^2 x^2\right )+8 c^5 d^5 x^5\right )}{(d+e x)^6 \left (a e^2+c d^2\right )^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(5/2)/(d + e*x)^7,x]

[Out]

((Sqrt[a + c*x^2]*(-8*a^5*e^5 + 8*c^5*d^5*x^5 - 2*a^4*c*e^3*(13*d^2 + 6*d*e*x +
13*e^2*x^2) + 2*a*c^4*d^3*x^3*(13*d^2 + 6*d*e*x + 13*e^2*x^2) - a^3*c^2*e*(33*d^
4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4) + a^2*c^3*d*x*(33*
d^4 + 54*d^3*e*x + 122*d^2*e^2*x^2 + 54*d*e^3*x^3 + 33*e^4*x^4)))/((c*d^2 + a*e^
2)^3*(d + e*x)^6) + (15*a^3*c^3*Log[d + e*x])/(c*d^2 + a*e^2)^(7/2) - (15*a^3*c^
3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(7/2))
/48

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Maple [B]  time = 0.049, size = 7616, normalized size = 37.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(5/2)/(e*x+d)^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)^(5/2)/(e*x + d)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.37852, 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)^7,x, algorithm="fricas")

[Out]

[-1/96*(2*(33*a^3*c^2*d^4*e + 26*a^4*c*d^2*e^3 + 8*a^5*e^5 - (8*c^5*d^5 + 26*a*c
^4*d^3*e^2 + 33*a^2*c^3*d*e^4)*x^5 - 3*(4*a*c^4*d^4*e + 18*a^2*c^3*d^2*e^3 - 11*
a^3*c^2*e^5)*x^4 - 2*(13*a*c^4*d^5 + 61*a^2*c^3*d^3*e^2 - 27*a^3*c^2*d*e^4)*x^3
- 2*(27*a^2*c^3*d^4*e - 61*a^3*c^2*d^2*e^3 - 13*a^4*c*e^5)*x^2 - 3*(11*a^2*c^3*d
^5 - 18*a^3*c^2*d^3*e^2 - 4*a^4*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a)
- 15*(a^3*c^3*e^6*x^6 + 6*a^3*c^3*d*e^5*x^5 + 15*a^3*c^3*d^2*e^4*x^4 + 20*a^3*c^
3*d^3*e^3*x^3 + 15*a^3*c^3*d^4*e^2*x^2 + 6*a^3*c^3*d^5*e*x + a^3*c^3*d^6)*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^3*d^12 + 3*a*c^2*d^10*e^2 + 3*a^2*c*d^8*e^4 + a^3*d^6*e^
6 + (c^3*d^6*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12)*x^6 + 6*(c^3*d
^7*e^5 + 3*a*c^2*d^5*e^7 + 3*a^2*c*d^3*e^9 + a^3*d*e^11)*x^5 + 15*(c^3*d^8*e^4 +
3*a*c^2*d^6*e^6 + 3*a^2*c*d^4*e^8 + a^3*d^2*e^10)*x^4 + 20*(c^3*d^9*e^3 + 3*a*c
^2*d^7*e^5 + 3*a^2*c*d^5*e^7 + a^3*d^3*e^9)*x^3 + 15*(c^3*d^10*e^2 + 3*a*c^2*d^8
*e^4 + 3*a^2*c*d^6*e^6 + a^3*d^4*e^8)*x^2 + 6*(c^3*d^11*e + 3*a*c^2*d^9*e^3 + 3*
a^2*c*d^7*e^5 + a^3*d^5*e^7)*x)*sqrt(c*d^2 + a*e^2)), -1/48*((33*a^3*c^2*d^4*e +
26*a^4*c*d^2*e^3 + 8*a^5*e^5 - (8*c^5*d^5 + 26*a*c^4*d^3*e^2 + 33*a^2*c^3*d*e^4
)*x^5 - 3*(4*a*c^4*d^4*e + 18*a^2*c^3*d^2*e^3 - 11*a^3*c^2*e^5)*x^4 - 2*(13*a*c^
4*d^5 + 61*a^2*c^3*d^3*e^2 - 27*a^3*c^2*d*e^4)*x^3 - 2*(27*a^2*c^3*d^4*e - 61*a^
3*c^2*d^2*e^3 - 13*a^4*c*e^5)*x^2 - 3*(11*a^2*c^3*d^5 - 18*a^3*c^2*d^3*e^2 - 4*a
^4*c*d*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 15*(a^3*c^3*e^6*x^6 + 6*a^
3*c^3*d*e^5*x^5 + 15*a^3*c^3*d^2*e^4*x^4 + 20*a^3*c^3*d^3*e^3*x^3 + 15*a^3*c^3*d
^4*e^2*x^2 + 6*a^3*c^3*d^5*e*x + a^3*c^3*d^6)*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^3*d^12 + 3*a*c^2*d^10*e^2 + 3*a^
2*c*d^8*e^4 + a^3*d^6*e^6 + (c^3*d^6*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 +
a^3*e^12)*x^6 + 6*(c^3*d^7*e^5 + 3*a*c^2*d^5*e^7 + 3*a^2*c*d^3*e^9 + a^3*d*e^11)
*x^5 + 15*(c^3*d^8*e^4 + 3*a*c^2*d^6*e^6 + 3*a^2*c*d^4*e^8 + a^3*d^2*e^10)*x^4 +
20*(c^3*d^9*e^3 + 3*a*c^2*d^7*e^5 + 3*a^2*c*d^5*e^7 + a^3*d^3*e^9)*x^3 + 15*(c^
3*d^10*e^2 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^6*e^6 + a^3*d^4*e^8)*x^2 + 6*(c^3*d^11*
e + 3*a*c^2*d^9*e^3 + 3*a^2*c*d^7*e^5 + a^3*d^5*e^7)*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)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.325117, 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)^7,x, algorithm="giac")

[Out]

Done