3.531 \(\int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=153 \[ -\frac{3 a^2 c^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{5/2}}-\frac{3 a c \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

[Out]

(-3*a*c*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^2*(d + e*x)^2) - ((a*e
- c*d*x)*(a + c*x^2)^(3/2))/(4*(c*d^2 + a*e^2)*(d + e*x)^4) - (3*a^2*c^2*ArcTan
h[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(5/2)
)

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Rubi [A]  time = 0.206628, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{3 a^2 c^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{5/2}}-\frac{3 a c \sqrt{a+c x^2} (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{\left (a+c x^2\right )^{3/2} (a e-c d x)}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

(-3*a*c*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^2*(d + e*x)^2) - ((a*e
- c*d*x)*(a + c*x^2)^(3/2))/(4*(c*d^2 + a*e^2)*(d + e*x)^4) - (3*a^2*c^2*ArcTan
h[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(5/2)
)

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Rubi in Sympy [A]  time = 23.7087, size = 146, normalized size = 0.95 \[ - \frac{3 a^{2} c^{2} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{8 \left (a e^{2} + c d^{2}\right )^{\frac{5}{2}}} - \frac{3 a c \sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right )}{16 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (2 a e - 2 c d x\right )}{8 \left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(3/2)/(e*x+d)**5,x)

[Out]

-3*a**2*c**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)**(5/2)) - 3*a*c*sqrt(a + c*x**2)*(2*a*e - 2*c*d*x)/(16*(d + e*x)
**2*(a*e**2 + c*d**2)**2) - (a + c*x**2)**(3/2)*(2*a*e - 2*c*d*x)/(8*(d + e*x)**
4*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.463291, size = 198, normalized size = 1.29 \[ \frac{1}{8} \left (-\frac{3 a^2 c^2 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{5/2}}+\frac{3 a^2 c^2 \log (d+e x)}{\left (a e^2+c d^2\right )^{5/2}}+\frac{\sqrt{a+c x^2} \left (-2 a^3 e^3-a^2 c e \left (5 d^2+4 d e x+5 e^2 x^2\right )+a c^2 d x \left (5 d^2+4 d e x+5 e^2 x^2\right )+2 c^3 d^3 x^3\right )}{(d+e x)^4 \left (a e^2+c d^2\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(3/2)/(d + e*x)^5,x]

[Out]

((Sqrt[a + c*x^2]*(-2*a^3*e^3 + 2*c^3*d^3*x^3 - a^2*c*e*(5*d^2 + 4*d*e*x + 5*e^2
*x^2) + a*c^2*d*x*(5*d^2 + 4*d*e*x + 5*e^2*x^2)))/((c*d^2 + a*e^2)^2*(d + e*x)^4
) + (3*a^2*c^2*Log[d + e*x])/(c*d^2 + a*e^2)^(5/2) - (3*a^2*c^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)^(5/2))/8

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Maple [B]  time = 0.025, size = 3528, normalized size = 23.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(3/2)/(e*x+d)^5,x)

[Out]

-3/8/e^4/(a*e^2+c*d^2)^2*c^(7/2)*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))+1/8/e*c^4*d^4/(a*e^2+c*d^2)^4*(c*(d/e+
x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)+3/8/e^3*c^5*d^6/(a*e^2+c*d^2)^4*(c
*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-1/4/e*c^3*d^2/(a*e^2+c*d^2)^
3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-3/4/e^3*c^4*d^4/(a*e^2+c
*d^2)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)-3/4/e^3*c^5*d^6/(a
*e^2+c*d^2)^4/((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-3/8/e*c^4*d^4/(a*e^2+c*d^2)^4/((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+3/4/e*c^3*d^2/(a*e^2+c*d^2)^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+1/
8/e/(a*e^2+c*d^2)^2*c^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)-1/
4/e^3/(a*e^2+c*d^2)/(d/e+x)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5
/2)+3/2/e^3*c^4*d^4/(a*e^2+c*d^2)^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-3/4/e^3/(a*e^2+c*d^2)^2*c^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*d^2+1/8*c^3*d^3/
(a*e^2+c*d^2)^4/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)-1/
8*c^4*d^3/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*
x-3/8*c^2*d/(a*e^2+c*d^2)^3/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e
^2)^(5/2)+3/8*c^3*d/(a*e^2+c*d^2)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e
^2)^(3/2)*x-3/16*c^(7/2)*d^3/(a*e^2+c*d^2)^4*a^2*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))-3/8/e^4*c^(11/2)*d^7/(a*e^
2+c*d^2)^4*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))+3/4/e^4*c^(9/2)*d^5/(a*e^2+c*d^2)^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))-1/8/e/(a*e^2+c*d^2)^2*c
/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(5/2)+3/8/e/(a*e^2+c*
d^2)^2*c^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a+3/8/e^3/(a*e^
2+c*d^2)^2*c^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*d^2-3/16/e^
2*c^5*d^5/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*
x+3/8/e*c^4*d^4/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^
(1/2)*a-3/8/e^5*c^6*d^8/(a*e^2+c*d^2)^4/((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))+3/8/e^2*c^4*d^3/(a*e^2+c*d^2)^3*(c*(d/e+x)
^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/4/e*c^3*d^2/(a*e^2+c*d^2)^3*(c*(
d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*a+3/4/e^5*c^5*d^6/(a*e^2+c*d^2
)^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)
)-1/4/e^2*c*d/(a*e^2+c*d^2)^2/(d/e+x)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^
2)/e^2)^(5/2)-1/8/e*c^2*d^2/(a*e^2+c*d^2)^3/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+
x)+(a*e^2+c*d^2)/e^2)^(5/2)+9/16*c^3*d/(a*e^2+c*d^2)^3*a*(c*(d/e+x)^2-2*c*d/e*(d
/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/16*c^4*d^3/(a*e^2+c*d^2)^4*a*(c*(d/e+x)^2-2*c
*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/16/e^2/(a*e^2+c*d^2)^2*c^3*d*(c*(d/e+x
)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-3/8/e/(a*e^2+c*d^2)^2*c^2/((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^2-3/8/e
^5/(a*e^2+c*d^2)^2*c^4/((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))*d^4-9/16/e^2/(a*e^2+c*d^2)^2*c^(5/2)*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+9/8/e^2*c^(7/
2)*d^3/(a*e^2+c*d^2)^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-9/16/e^2*c^(9/2)*d^5/(a*e^2+c*d^2)^4*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+9/16*c
^(5/2)*d/(a*e^2+c*d^2)^3*a^2*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))

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.869226, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^5,x, algorithm="fricas")

[Out]

[-1/16*(2*(5*a^2*c*d^2*e + 2*a^3*e^3 - (2*c^3*d^3 + 5*a*c^2*d*e^2)*x^3 - (4*a*c^
2*d^2*e - 5*a^2*c*e^3)*x^2 - (5*a*c^2*d^3 - 4*a^2*c*d*e^2)*x)*sqrt(c*d^2 + a*e^2
)*sqrt(c*x^2 + a) - 3*(a^2*c^2*e^4*x^4 + 4*a^2*c^2*d*e^3*x^3 + 6*a^2*c^2*d^2*e^2
*x^2 + 4*a^2*c^2*d^3*e*x + a^2*c^2*d^4)*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^2*d^8 + 2*
a*c*d^6*e^2 + a^2*d^4*e^4 + (c^2*d^4*e^4 + 2*a*c*d^2*e^6 + a^2*e^8)*x^4 + 4*(c^2
*d^5*e^3 + 2*a*c*d^3*e^5 + a^2*d*e^7)*x^3 + 6*(c^2*d^6*e^2 + 2*a*c*d^4*e^4 + a^2
*d^2*e^6)*x^2 + 4*(c^2*d^7*e + 2*a*c*d^5*e^3 + a^2*d^3*e^5)*x)*sqrt(c*d^2 + a*e^
2)), -1/8*((5*a^2*c*d^2*e + 2*a^3*e^3 - (2*c^3*d^3 + 5*a*c^2*d*e^2)*x^3 - (4*a*c
^2*d^2*e - 5*a^2*c*e^3)*x^2 - (5*a*c^2*d^3 - 4*a^2*c*d*e^2)*x)*sqrt(-c*d^2 - a*e
^2)*sqrt(c*x^2 + a) - 3*(a^2*c^2*e^4*x^4 + 4*a^2*c^2*d*e^3*x^3 + 6*a^2*c^2*d^2*e
^2*x^2 + 4*a^2*c^2*d^3*e*x + a^2*c^2*d^4)*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^2*d^8 + 2*a*c*d^6*e^2 + a^2*d^4*e^4
+ (c^2*d^4*e^4 + 2*a*c*d^2*e^6 + a^2*e^8)*x^4 + 4*(c^2*d^5*e^3 + 2*a*c*d^3*e^5 +
a^2*d*e^7)*x^3 + 6*(c^2*d^6*e^2 + 2*a*c*d^4*e^4 + a^2*d^2*e^6)*x^2 + 4*(c^2*d^7
*e + 2*a*c*d^5*e^3 + a^2*d^3*e^5)*x)*sqrt(-c*d^2 - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(3/2)/(e*x+d)**5,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**5, x)

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GIAC/XCAS [A]  time = 1.22065, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^5,x, algorithm="giac")

[Out]

Done