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

Optimal. Leaf size=300 \[ \frac{a^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{128 c^{9/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{192 c^3 d \left (c+d x^2\right )^2 (b c-a d)}+\frac{x \sqrt{a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{384 c^4 d \left (c+d x^2\right ) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (7 a d+2 b c)}{48 c^2 d \left (c+d x^2\right )^3}-\frac{x \sqrt{a+b x^2} (b c-a d)}{8 c d \left (c+d x^2\right )^4} \]

[Out]

-((b*c - a*d)*x*Sqrt[a + b*x^2])/(8*c*d*(c + d*x^2)^4) + ((2*b*c + 7*a*d)*x*Sqrt
[a + b*x^2])/(48*c^2*d*(c + d*x^2)^3) + ((8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x
*Sqrt[a + b*x^2])/(192*c^3*d*(b*c - a*d)*(c + d*x^2)^2) + ((16*b^3*c^3 + 40*a*b^
2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(384*c^4*d*(b*c - a*
d)^2*(c + d*x^2)) + (a^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b*
c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(128*c^(9/2)*(b*c - a*d)^(5/2))

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Rubi [A]  time = 0.91508, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{a^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \tanh ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{128 c^{9/2} (b c-a d)^{5/2}}+\frac{x \sqrt{a+b x^2} \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right )}{192 c^3 d \left (c+d x^2\right )^2 (b c-a d)}+\frac{x \sqrt{a+b x^2} \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )}{384 c^4 d \left (c+d x^2\right ) (b c-a d)^2}+\frac{x \sqrt{a+b x^2} (7 a d+2 b c)}{48 c^2 d \left (c+d x^2\right )^3}-\frac{x \sqrt{a+b x^2} (b c-a d)}{8 c d \left (c+d x^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x*Sqrt[a + b*x^2])/(8*c*d*(c + d*x^2)^4) + ((2*b*c + 7*a*d)*x*Sqrt
[a + b*x^2])/(48*c^2*d*(c + d*x^2)^3) + ((8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*x
*Sqrt[a + b*x^2])/(192*c^3*d*(b*c - a*d)*(c + d*x^2)^2) + ((16*b^3*c^3 + 40*a*b^
2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(384*c^4*d*(b*c - a*
d)^2*(c + d*x^2)) + (a^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b*
c - a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/(128*c^(9/2)*(b*c - a*d)^(5/2))

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Rubi in Sympy [A]  time = 158.453, size = 277, normalized size = 0.92 \[ \frac{a^{2} \left (35 a^{2} d^{2} - 80 a b c d + 48 b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{x \sqrt{a d - b c}}{\sqrt{c} \sqrt{a + b x^{2}}} \right )}}{128 c^{\frac{9}{2}} \left (a d - b c\right )^{\frac{5}{2}}} + \frac{x \sqrt{a + b x^{2}} \left (a d - b c\right )}{8 c d \left (c + d x^{2}\right )^{4}} + \frac{x \sqrt{a + b x^{2}} \left (7 a d + 2 b c\right )}{48 c^{2} d \left (c + d x^{2}\right )^{3}} + \frac{x \sqrt{a + b x^{2}} \left (35 a^{2} d^{2} - 24 a b c d - 8 b^{2} c^{2}\right )}{192 c^{3} d \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} + \frac{x \sqrt{a + b x^{2}} \left (105 a^{3} d^{3} - 170 a^{2} b c d^{2} + 40 a b^{2} c^{2} d + 16 b^{3} c^{3}\right )}{384 c^{4} d \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(35*a**2*d**2 - 80*a*b*c*d + 48*b**2*c**2)*atan(x*sqrt(a*d - b*c)/(sqrt(c)*
sqrt(a + b*x**2)))/(128*c**(9/2)*(a*d - b*c)**(5/2)) + x*sqrt(a + b*x**2)*(a*d -
b*c)/(8*c*d*(c + d*x**2)**4) + x*sqrt(a + b*x**2)*(7*a*d + 2*b*c)/(48*c**2*d*(c
+ d*x**2)**3) + x*sqrt(a + b*x**2)*(35*a**2*d**2 - 24*a*b*c*d - 8*b**2*c**2)/(1
92*c**3*d*(c + d*x**2)**2*(a*d - b*c)) + x*sqrt(a + b*x**2)*(105*a**3*d**3 - 170
*a**2*b*c*d**2 + 40*a*b**2*c**2*d + 16*b**3*c**3)/(384*c**4*d*(c + d*x**2)*(a*d
- b*c)**2)

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Mathematica [A]  time = 0.452451, size = 260, normalized size = 0.87 \[ \frac{\frac{3 a^2 \left (35 a^2 d^2-80 a b c d+48 b^2 c^2\right ) \tan ^{-1}\left (\frac{x \sqrt{a d-b c}}{\sqrt{c} \sqrt{a+b x^2}}\right )}{\sqrt{a d-b c}}-\frac{\sqrt{c} x \sqrt{a+b x^2} \left (-2 c \left (c+d x^2\right )^2 \left (-35 a^2 d^2+24 a b c d+8 b^2 c^2\right ) (b c-a d)-\left (c+d x^2\right )^3 \left (105 a^3 d^3-170 a^2 b c d^2+40 a b^2 c^2 d+16 b^3 c^3\right )+48 c^3 (b c-a d)^3-8 c^2 \left (c+d x^2\right ) (7 a d+2 b c) (b c-a d)^2\right )}{d \left (c+d x^2\right )^4}}{384 c^{9/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-((Sqrt[c]*x*Sqrt[a + b*x^2]*(48*c^3*(b*c - a*d)^3 - 8*c^2*(b*c - a*d)^2*(2*b*c
+ 7*a*d)*(c + d*x^2) - 2*c*(b*c - a*d)*(8*b^2*c^2 + 24*a*b*c*d - 35*a^2*d^2)*(c
+ d*x^2)^2 - (16*b^3*c^3 + 40*a*b^2*c^2*d - 170*a^2*b*c*d^2 + 105*a^3*d^3)*(c +
d*x^2)^3))/(d*(c + d*x^2)^4)) + (3*a^2*(48*b^2*c^2 - 80*a*b*c*d + 35*a^2*d^2)*A
rcTan[(Sqrt[-(b*c) + a*d]*x)/(Sqrt[c]*Sqrt[a + b*x^2])])/Sqrt[-(b*c) + a*d])/(38
4*c^(9/2)*(b*c - a*d)^2)

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Maple [B]  time = 0.075, size = 18791, normalized size = 62.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/2)/(d*x^2 + c)^5, x)

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Fricas [A]  time = 2.4559, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1536*(4*((16*b^3*c^3*d^2 + 40*a*b^2*c^2*d^3 - 170*a^2*b*c*d^4 + 105*a^3*d^5)*
x^7 + (64*b^3*c^4*d + 152*a*b^2*c^3*d^2 - 628*a^2*b*c^2*d^3 + 385*a^3*c*d^4)*x^5
+ (96*b^3*c^5 + 208*a*b^2*c^4*d - 842*a^2*b*c^3*d^2 + 511*a^3*c^2*d^3)*x^3 + 3*
(80*a*b^2*c^5 - 176*a^2*b*c^4*d + 93*a^3*c^3*d^2)*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*
x^2 + a) + 3*(48*a^2*b^2*c^6 - 80*a^3*b*c^5*d + 35*a^4*c^4*d^2 + (48*a^2*b^2*c^2
*d^4 - 80*a^3*b*c*d^5 + 35*a^4*d^6)*x^8 + 4*(48*a^2*b^2*c^3*d^3 - 80*a^3*b*c^2*d
^4 + 35*a^4*c*d^5)*x^6 + 6*(48*a^2*b^2*c^4*d^2 - 80*a^3*b*c^3*d^3 + 35*a^4*c^2*d
^4)*x^4 + 4*(48*a^2*b^2*c^5*d - 80*a^3*b*c^4*d^2 + 35*a^4*c^3*d^3)*x^2)*log((((8
*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*x^4 + a^2*c^2 + 2*(4*a*b*c^2 - 3*a^2*c*d)*x^2)*s
qrt(b*c^2 - a*c*d) + 4*((2*b^2*c^3 - 3*a*b*c^2*d + a^2*c*d^2)*x^3 + (a*b*c^3 - a
^2*c^2*d)*x)*sqrt(b*x^2 + a))/(d^2*x^4 + 2*c*d*x^2 + c^2)))/((b^2*c^10 - 2*a*b*c
^9*d + a^2*c^8*d^2 + (b^2*c^6*d^4 - 2*a*b*c^5*d^5 + a^2*c^4*d^6)*x^8 + 4*(b^2*c^
7*d^3 - 2*a*b*c^6*d^4 + a^2*c^5*d^5)*x^6 + 6*(b^2*c^8*d^2 - 2*a*b*c^7*d^3 + a^2*
c^6*d^4)*x^4 + 4*(b^2*c^9*d - 2*a*b*c^8*d^2 + a^2*c^7*d^3)*x^2)*sqrt(b*c^2 - a*c
*d)), 1/768*(2*((16*b^3*c^3*d^2 + 40*a*b^2*c^2*d^3 - 170*a^2*b*c*d^4 + 105*a^3*d
^5)*x^7 + (64*b^3*c^4*d + 152*a*b^2*c^3*d^2 - 628*a^2*b*c^2*d^3 + 385*a^3*c*d^4)
*x^5 + (96*b^3*c^5 + 208*a*b^2*c^4*d - 842*a^2*b*c^3*d^2 + 511*a^3*c^2*d^3)*x^3
+ 3*(80*a*b^2*c^5 - 176*a^2*b*c^4*d + 93*a^3*c^3*d^2)*x)*sqrt(-b*c^2 + a*c*d)*sq
rt(b*x^2 + a) + 3*(48*a^2*b^2*c^6 - 80*a^3*b*c^5*d + 35*a^4*c^4*d^2 + (48*a^2*b^
2*c^2*d^4 - 80*a^3*b*c*d^5 + 35*a^4*d^6)*x^8 + 4*(48*a^2*b^2*c^3*d^3 - 80*a^3*b*
c^2*d^4 + 35*a^4*c*d^5)*x^6 + 6*(48*a^2*b^2*c^4*d^2 - 80*a^3*b*c^3*d^3 + 35*a^4*
c^2*d^4)*x^4 + 4*(48*a^2*b^2*c^5*d - 80*a^3*b*c^4*d^2 + 35*a^4*c^3*d^3)*x^2)*arc
tan(1/2*sqrt(-b*c^2 + a*c*d)*((2*b*c - a*d)*x^2 + a*c)/((b*c^2 - a*c*d)*sqrt(b*x
^2 + a)*x)))/((b^2*c^10 - 2*a*b*c^9*d + a^2*c^8*d^2 + (b^2*c^6*d^4 - 2*a*b*c^5*d
^5 + a^2*c^4*d^6)*x^8 + 4*(b^2*c^7*d^3 - 2*a*b*c^6*d^4 + a^2*c^5*d^5)*x^6 + 6*(b
^2*c^8*d^2 - 2*a*b*c^7*d^3 + a^2*c^6*d^4)*x^4 + 4*(b^2*c^9*d - 2*a*b*c^8*d^2 + a
^2*c^7*d^3)*x^2)*sqrt(-b*c^2 + a*c*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 1.5878, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x