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

Optimal. Leaf size=198 \[ -\frac{(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{9/2}}+\frac{\sqrt{c+d x^2} (2 b c-7 a d) (b c-a d)}{2 b^4}+\frac{\left (c+d x^2\right )^{3/2} (2 b c-7 a d)}{6 b^3}+\frac{\left (c+d x^2\right )^{5/2} (2 b c-7 a d)}{10 b^2 (b c-a d)}+\frac{a \left (c+d x^2\right )^{7/2}}{2 b \left (a+b x^2\right ) (b c-a d)} \]

[Out]

((2*b*c - 7*a*d)*(b*c - a*d)*Sqrt[c + d*x^2])/(2*b^4) + ((2*b*c - 7*a*d)*(c + d*
x^2)^(3/2))/(6*b^3) + ((2*b*c - 7*a*d)*(c + d*x^2)^(5/2))/(10*b^2*(b*c - a*d)) +
(a*(c + d*x^2)^(7/2))/(2*b*(b*c - a*d)*(a + b*x^2)) - ((2*b*c - 7*a*d)*(b*c - a
*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*b^(9/2))

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Rubi [A]  time = 0.463654, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{(2 b c-7 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{2 b^{9/2}}+\frac{\sqrt{c+d x^2} (2 b c-7 a d) (b c-a d)}{2 b^4}+\frac{\left (c+d x^2\right )^{3/2} (2 b c-7 a d)}{6 b^3}+\frac{\left (c+d x^2\right )^{5/2} (2 b c-7 a d)}{10 b^2 (b c-a d)}+\frac{a \left (c+d x^2\right )^{7/2}}{2 b \left (a+b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

((2*b*c - 7*a*d)*(b*c - a*d)*Sqrt[c + d*x^2])/(2*b^4) + ((2*b*c - 7*a*d)*(c + d*
x^2)^(3/2))/(6*b^3) + ((2*b*c - 7*a*d)*(c + d*x^2)^(5/2))/(10*b^2*(b*c - a*d)) +
(a*(c + d*x^2)^(7/2))/(2*b*(b*c - a*d)*(a + b*x^2)) - ((2*b*c - 7*a*d)*(b*c - a
*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(2*b^(9/2))

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Rubi in Sympy [A]  time = 53.1518, size = 170, normalized size = 0.86 \[ - \frac{a \left (c + d x^{2}\right )^{\frac{7}{2}}}{2 b \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{\left (c + d x^{2}\right )^{\frac{5}{2}} \left (\frac{7 a d}{2} - b c\right )}{5 b^{2} \left (a d - b c\right )} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (\frac{7 a d}{2} - b c\right )}{3 b^{3}} + \frac{\sqrt{c + d x^{2}} \left (a d - b c\right ) \left (7 a d - 2 b c\right )}{2 b^{4}} - \frac{\left (a d - b c\right )^{\frac{3}{2}} \left (\frac{7 a d}{2} - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(c + d*x**2)**(7/2)/(2*b*(a + b*x**2)*(a*d - b*c)) + (c + d*x**2)**(5/2)*(7*a
*d/2 - b*c)/(5*b**2*(a*d - b*c)) - (c + d*x**2)**(3/2)*(7*a*d/2 - b*c)/(3*b**3)
+ sqrt(c + d*x**2)*(a*d - b*c)*(7*a*d - 2*b*c)/(2*b**4) - (a*d - b*c)**(3/2)*(7*
a*d/2 - b*c)*atan(sqrt(b)*sqrt(c + d*x**2)/sqrt(a*d - b*c))/b**(9/2)

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Mathematica [C]  time = 0.92877, size = 332, normalized size = 1.68 \[ \frac{2 \sqrt{b} \sqrt{c+d x^2} \left (90 a^2 d^2+2 b d x^2 (11 b c-10 a d)+\frac{15 a (b c-a d)^2}{a+b x^2}-140 a b c d+46 b^2 c^2+6 b^2 d^2 x^4\right )-15 (2 b c-7 a d) (b c-a d)^{3/2} \log \left (\frac{4 b^{9/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) (2 b c-7 a d) (b c-a d)^{5/2}}\right )-15 (2 b c-7 a d) (b c-a d)^{3/2} \log \left (\frac{4 b^{9/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) (2 b c-7 a d) (b c-a d)^{5/2}}\right )}{60 b^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b]*Sqrt[c + d*x^2]*(46*b^2*c^2 - 140*a*b*c*d + 90*a^2*d^2 + 2*b*d*(11*b*
c - 10*a*d)*x^2 + 6*b^2*d^2*x^4 + (15*a*(b*c - a*d)^2)/(a + b*x^2)) - 15*(2*b*c
- 7*a*d)*(b*c - a*d)^(3/2)*Log[(4*b^(9/2)*(Sqrt[b]*c - I*Sqrt[a]*d*x + Sqrt[b*c
- a*d]*Sqrt[c + d*x^2]))/((2*b*c - 7*a*d)*(b*c - a*d)^(5/2)*(I*Sqrt[a] + Sqrt[b]
*x))] - 15*(2*b*c - 7*a*d)*(b*c - a*d)^(3/2)*Log[(4*b^(9/2)*(Sqrt[b]*c + I*Sqrt[
a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/((2*b*c - 7*a*d)*(b*c - a*d)^(5/2)*((
-I)*Sqrt[a] + Sqrt[b]*x))])/(60*b^(9/2))

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Maple [B]  time = 0.027, size = 7443, normalized size = 37.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.304577, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (2 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 7 \, a^{3} d^{2} +{\left (2 \, b^{3} c^{2} - 9 \, a b^{2} c d + 7 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left (6 \, b^{3} d^{2} x^{6} + 61 \, a b^{2} c^{2} - 170 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (11 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{4} + 2 \,{\left (23 \, b^{3} c^{2} - 59 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{120 \,{\left (b^{5} x^{2} + a b^{4}\right )}}, -\frac{15 \,{\left (2 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 7 \, a^{3} d^{2} +{\left (2 \, b^{3} c^{2} - 9 \, a b^{2} c d + 7 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) - 2 \,{\left (6 \, b^{3} d^{2} x^{6} + 61 \, a b^{2} c^{2} - 170 \, a^{2} b c d + 105 \, a^{3} d^{2} + 2 \,{\left (11 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{4} + 2 \,{\left (23 \, b^{3} c^{2} - 59 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{60 \,{\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/120*(15*(2*a*b^2*c^2 - 9*a^2*b*c*d + 7*a^3*d^2 + (2*b^3*c^2 - 9*a*b^2*c*d + 7
*a^2*b*d^2)*x^2)*sqrt((b*c - a*d)/b)*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d +
a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d
*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 4*(6*b^3*d^2*x^6 +
61*a*b^2*c^2 - 170*a^2*b*c*d + 105*a^3*d^2 + 2*(11*b^3*c*d - 7*a*b^2*d^2)*x^4 +
2*(23*b^3*c^2 - 59*a*b^2*c*d + 35*a^2*b*d^2)*x^2)*sqrt(d*x^2 + c))/(b^5*x^2 + a
*b^4), -1/60*(15*(2*a*b^2*c^2 - 9*a^2*b*c*d + 7*a^3*d^2 + (2*b^3*c^2 - 9*a*b^2*c
*d + 7*a^2*b*d^2)*x^2)*sqrt(-(b*c - a*d)/b)*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)/(
sqrt(d*x^2 + c)*b*sqrt(-(b*c - a*d)/b))) - 2*(6*b^3*d^2*x^6 + 61*a*b^2*c^2 - 170
*a^2*b*c*d + 105*a^3*d^2 + 2*(11*b^3*c*d - 7*a*b^2*d^2)*x^4 + 2*(23*b^3*c^2 - 59
*a*b^2*c*d + 35*a^2*b*d^2)*x^2)*sqrt(d*x^2 + c))/(b^5*x^2 + a*b^4)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.22858, size = 356, normalized size = 1.8 \[ \frac{{\left (2 \, b^{3} c^{3} - 11 \, a b^{2} c^{2} d + 16 \, a^{2} b c d^{2} - 7 \, a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{2 \, \sqrt{-b^{2} c + a b d} b^{4}} + \frac{\sqrt{d x^{2} + c} a b^{2} c^{2} d - 2 \, \sqrt{d x^{2} + c} a^{2} b c d^{2} + \sqrt{d x^{2} + c} a^{3} d^{3}}{2 \,{\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} b^{4}} + \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{8} + 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{8} c + 15 \, \sqrt{d x^{2} + c} b^{8} c^{2} - 10 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b^{7} d - 60 \, \sqrt{d x^{2} + c} a b^{7} c d + 45 \, \sqrt{d x^{2} + c} a^{2} b^{6} d^{2}}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b^3*c^3 - 11*a*b^2*c^2*d + 16*a^2*b*c*d^2 - 7*a^3*d^3)*arctan(sqrt(d*x^2
+ c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^4) + 1/2*(sqrt(d*x^2 + c)*a
*b^2*c^2*d - 2*sqrt(d*x^2 + c)*a^2*b*c*d^2 + sqrt(d*x^2 + c)*a^3*d^3)/(((d*x^2 +
c)*b - b*c + a*d)*b^4) + 1/15*(3*(d*x^2 + c)^(5/2)*b^8 + 5*(d*x^2 + c)^(3/2)*b^
8*c + 15*sqrt(d*x^2 + c)*b^8*c^2 - 10*(d*x^2 + c)^(3/2)*a*b^7*d - 60*sqrt(d*x^2
+ c)*a*b^7*c*d + 45*sqrt(d*x^2 + c)*a^2*b^6*d^2)/b^10