3.697 \(\int \frac{x \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx\)
Optimal. Leaf size=119 \[ -\frac{(b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{7/2}}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{b^3}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{3 b^2}+\frac{\left (c+d x^2\right )^{5/2}}{5 b} \]
[Out]
((b*c - a*d)^2*Sqrt[c + d*x^2])/b^3 + ((b*c - a*d)*(c + d*x^2)^(3/2))/(3*b^2) +
(c + d*x^2)^(5/2)/(5*b) - ((b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/S
qrt[b*c - a*d]])/b^(7/2)
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Rubi [A] time = 0.25741, antiderivative size = 119, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182
\[ -\frac{(b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{7/2}}+\frac{\sqrt{c+d x^2} (b c-a d)^2}{b^3}+\frac{\left (c+d x^2\right )^{3/2} (b c-a d)}{3 b^2}+\frac{\left (c+d x^2\right )^{5/2}}{5 b} \]
Antiderivative was successfully verified.
[In] Int[(x*(c + d*x^2)^(5/2))/(a + b*x^2),x]
[Out]
((b*c - a*d)^2*Sqrt[c + d*x^2])/b^3 + ((b*c - a*d)*(c + d*x^2)^(3/2))/(3*b^2) +
(c + d*x^2)^(5/2)/(5*b) - ((b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/S
qrt[b*c - a*d]])/b^(7/2)
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Rubi in Sympy [A] time = 34.9982, size = 99, normalized size = 0.83 \[ \frac{\left (c + d x^{2}\right )^{\frac{5}{2}}}{5 b} - \frac{\left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 b^{2}} + \frac{\sqrt{c + d x^{2}} \left (a d - b c\right )^{2}}{b^{3}} - \frac{\left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x**2+c)**(5/2)/(b*x**2+a),x)
[Out]
(c + d*x**2)**(5/2)/(5*b) - (c + d*x**2)**(3/2)*(a*d - b*c)/(3*b**2) + sqrt(c +
d*x**2)*(a*d - b*c)**2/b**3 - (a*d - b*c)**(5/2)*atan(sqrt(b)*sqrt(c + d*x**2)/s
qrt(a*d - b*c))/b**(7/2)
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Mathematica [C] time = 0.570494, size = 268, normalized size = 2.25 \[ \frac{2 \sqrt{b} \sqrt{c+d x^2} \left (15 a^2 d^2-5 a b d \left (7 c+d x^2\right )+b^2 \left (23 c^2+11 c d x^2+3 d^2 x^4\right )\right )-15 (b c-a d)^{5/2} \log \left (\frac{2 b^{7/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{7/2}}\right )-15 (b c-a d)^{5/2} \log \left (\frac{2 b^{7/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{7/2}}\right )}{30 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(c + d*x^2)^(5/2))/(a + b*x^2),x]
[Out]
(2*Sqrt[b]*Sqrt[c + d*x^2]*(15*a^2*d^2 - 5*a*b*d*(7*c + d*x^2) + b^2*(23*c^2 + 1
1*c*d*x^2 + 3*d^2*x^4)) - 15*(b*c - a*d)^(5/2)*Log[(2*b^(7/2)*(Sqrt[b]*c - I*Sqr
t[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/((b*c - a*d)^(7/2)*(I*Sqrt[a] + Sqr
t[b]*x))] - 15*(b*c - a*d)^(5/2)*Log[(2*b^(7/2)*(Sqrt[b]*c + I*Sqrt[a]*d*x + Sqr
t[b*c - a*d]*Sqrt[c + d*x^2]))/((b*c - a*d)^(7/2)*((-I)*Sqrt[a] + Sqrt[b]*x))])/
(30*b^(7/2))
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Maple [B] time = 0.019, size = 3078, normalized size = 25.9 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x^2+c)^(5/2)/(b*x^2+a),x)
[Out]
1/6/b*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c
)/b)^(3/2)*c+1/2/b*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1
/2))-(a*d-b*c)/b)^(1/2)*c^2+1/6/b*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(
x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*c+1/2/b*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a
*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*c^2+15/16/b^2*d^(1/2)*(-a*b)
^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2)
)^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c^2-1/b^2*((x-
1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)
*a*d*c-15/16/b^2*d^(1/2)*(-a*b)^(1/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))
*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a
*d-b*c)/b)^(1/2))*c^2-1/b^2*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*
(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*a*d*c-1/2/b^4*d^(5/2)*(-a*b)^(1/2)*ln((-d*(-a*b
)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(
1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*a^2+1/2/b^4/(-(a*d-b*c)/b)^(1/2)
*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/
2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b
)^(1/2))/(x+1/b*(-a*b)^(1/2)))*a^3*d^3+1/2/b^4*d^(5/2)*(-a*b)^(1/2)*ln((d*(-a*b)
^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*a^2+1/2/b^4/(-(a*d-b*c)/b)^(1/2)*
ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2
)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)
^(1/2))/(x-1/b*(-a*b)^(1/2)))*a^3*d^3+5/4/b^3*d^(3/2)*(-a*b)^(1/2)*ln((-d*(-a*b)
^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1
/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*a*c-3/2/b^3/(-(a*d-b*c)/b)^(1/2)*
ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2
)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)
^(1/2))/(x+1/b*(-a*b)^(1/2)))*a^2*d^2*c-5/4/b^3*d^(3/2)*(-a*b)^(1/2)*ln((d*(-a*b
)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(
1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*a*c-3/2/b^3/(-(a*d-b*c)/b)^(1/2)
*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/
2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b
)^(1/2))/(x-1/b*(-a*b)^(1/2)))*a^2*d^2*c+3/2/b^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*
d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*
(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x
-1/b*(-a*b)^(1/2)))*a*d*c^2+7/16/b^2*d*(-a*b)^(1/2)*c*((x-1/b*(-a*b)^(1/2))^2*d+
2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-1/4/b^3*d^2*(-a*b)^
(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c
)/b)^(1/2)*x*a+1/8/b^2*d*(-a*b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)
/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*x+1/10/b*((x+1/b*(-a*b)^(1/2))^2*d-2*
d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)+1/10/b*((x-1/b*(-a*b)^(
1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)-1/6/b^2*((x
+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2
)*a*d+1/2/b^3*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-
(a*d-b*c)/b)^(1/2)*a^2*d^2-1/2/b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a
*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*
d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2
)))*c^3+3/2/b^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/
b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2
)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*a*d*c^2-7/16/
b^2*d*(-a*b)^(1/2)*c*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^
(1/2))-(a*d-b*c)/b)^(1/2)*x+1/4/b^3*d^2*(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2
*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x*a-1/8/b^2*d*(-a*b)^(
1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)
/b)^(3/2)*x-1/2/b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-
1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*c^3-1/6/b^2
*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^
(3/2)*a*d+1/2/b^3*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/
2))-(a*d-b*c)/b)^(1/2)*a^2*d^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((d*x^2 + c)^(5/2)*x/(b*x^2 + a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.286611, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left (3 \, b^{2} d^{2} x^{4} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} +{\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{60 \, b^{3}}, -\frac{15 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) - 2 \,{\left (3 \, b^{2} d^{2} x^{4} + 23 \, b^{2} c^{2} - 35 \, a b c d + 15 \, a^{2} d^{2} +{\left (11 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{30 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x/(b*x^2 + a),x, algorithm="fricas")
[Out]
[1/60*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^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*(3*b^2*d^2*x^4 + 23*b^2*c^2 - 35*a*b*c*d + 15*a^2*d^2 + (11*b^2*c*d - 5
*a*b*d^2)*x^2)*sqrt(d*x^2 + c))/b^3, -1/30*(15*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*s
qrt(-(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*(3*b^2*d^2*x^4 + 23*b^2*c^2 - 35*a*b*c*d + 15*a^2*d^2 + (11
*b^2*c*d - 5*a*b*d^2)*x^2)*sqrt(d*x^2 + c))/b^3]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (c + d x^{2}\right )^{\frac{5}{2}}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x**2+c)**(5/2)/(b*x**2+a),x)
[Out]
Integral(x*(c + d*x**2)**(5/2)/(a + b*x**2), x)
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GIAC/XCAS [A] time = 0.236429, size = 248, normalized size = 2.08 \[ \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{3}} + \frac{3 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} b^{4} + 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} b^{4} c + 15 \, \sqrt{d x^{2} + c} b^{4} c^{2} - 5 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b^{3} d - 30 \, \sqrt{d x^{2} + c} a b^{3} c d + 15 \, \sqrt{d x^{2} + c} a^{2} b^{2} d^{2}}{15 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x/(b*x^2 + a),x, algorithm="giac")
[Out]
(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*arctan(sqrt(d*x^2 + c)*b/sqr
t(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^3) + 1/15*(3*(d*x^2 + c)^(5/2)*b^4 +
5*(d*x^2 + c)^(3/2)*b^4*c + 15*sqrt(d*x^2 + c)*b^4*c^2 - 5*(d*x^2 + c)^(3/2)*a*b
^3*d - 30*sqrt(d*x^2 + c)*a*b^3*c*d + 15*sqrt(d*x^2 + c)*a^2*b^2*d^2)/b^5