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

Optimal. Leaf size=237 \[ \frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{128 b^{3/2} d^{9/2}}-\frac{5 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2 (a d+7 b c)}{128 b d^4}+\frac{5 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d) (a d+7 b c)}{192 b d^3}-\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2} (a d+7 b c)}{48 b d^2}+\frac{\left (a+b x^2\right )^{7/2} \sqrt{c+d x^2}}{8 b d} \]

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(128*b*d^4) + (
5*(b*c - a*d)*(7*b*c + a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(192*b*d^3) - ((7
*b*c + a*d)*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(48*b*d^2) + ((a + b*x^2)^(7/2)*S
qrt[c + d*x^2])/(8*b*d) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(128*b^(3/2)*d^(9/2))

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Rubi [A]  time = 0.536146, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{5 (b c-a d)^3 (a d+7 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{128 b^{3/2} d^{9/2}}-\frac{5 \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2 (a d+7 b c)}{128 b d^4}+\frac{5 \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d) (a d+7 b c)}{192 b d^3}-\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2} (a d+7 b c)}{48 b d^2}+\frac{\left (a+b x^2\right )^{7/2} \sqrt{c+d x^2}}{8 b d} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(b*c - a*d)^2*(7*b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(128*b*d^4) + (
5*(b*c - a*d)*(7*b*c + a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(192*b*d^3) - ((7
*b*c + a*d)*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(48*b*d^2) + ((a + b*x^2)^(7/2)*S
qrt[c + d*x^2])/(8*b*d) + (5*(b*c - a*d)^3*(7*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a
+ b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(128*b^(3/2)*d^(9/2))

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Rubi in Sympy [A]  time = 54.3256, size = 212, normalized size = 0.89 \[ \frac{\left (a + b x^{2}\right )^{\frac{7}{2}} \sqrt{c + d x^{2}}}{8 b d} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \sqrt{c + d x^{2}} \left (a d + 7 b c\right )}{48 b d^{2}} - \frac{5 \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (a d + 7 b c\right )}{192 b d^{3}} - \frac{5 \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2} \left (a d + 7 b c\right )}{128 b d^{4}} - \frac{5 \left (a d - b c\right )^{3} \left (a d + 7 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{128 b^{\frac{3}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

(a + b*x**2)**(7/2)*sqrt(c + d*x**2)/(8*b*d) - (a + b*x**2)**(5/2)*sqrt(c + d*x*
*2)*(a*d + 7*b*c)/(48*b*d**2) - 5*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(a*d - b*
c)*(a*d + 7*b*c)/(192*b*d**3) - 5*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(a*d - b*c)*
*2*(a*d + 7*b*c)/(128*b*d**4) - 5*(a*d - b*c)**3*(a*d + 7*b*c)*atanh(sqrt(d)*sqr
t(a + b*x**2)/(sqrt(b)*sqrt(c + d*x**2)))/(128*b**(3/2)*d**(9/2))

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Mathematica [A]  time = 0.207506, size = 204, normalized size = 0.86 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (15 a^3 d^3+a^2 b d^2 \left (118 d x^2-191 c\right )+a b^2 d \left (265 c^2-172 c d x^2+136 d^2 x^4\right )+b^3 \left (-105 c^3+70 c^2 d x^2-56 c d^2 x^4+48 d^3 x^6\right )\right )}{384 b d^4}+\frac{5 (a d+7 b c) (b c-a d)^3 \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{256 b^{3/2} d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(15*a^3*d^3 + a^2*b*d^2*(-191*c + 118*d*x^2) +
a*b^2*d*(265*c^2 - 172*c*d*x^2 + 136*d^2*x^4) + b^3*(-105*c^3 + 70*c^2*d*x^2 - 5
6*c*d^2*x^4 + 48*d^3*x^6)))/(384*b*d^4) + (5*(b*c - a*d)^3*(7*b*c + a*d)*Log[b*c
+ a*d + 2*b*d*x^2 + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]])/(256*b^
(3/2)*d^(9/2))

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Maple [B]  time = 0.027, size = 770, normalized size = 3.3 \[ -{\frac{1}{768\,{d}^{4}b}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( -96\,{x}^{6}{b}^{3}{d}^{3}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}-272\,{x}^{4}a{b}^{2}{d}^{3}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+112\,{x}^{4}{b}^{3}c{d}^{2}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}-236\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}{a}^{2}b{d}^{3}\sqrt{bd}+344\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}ac{b}^{2}{d}^{2}\sqrt{bd}-140\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}{c}^{2}{b}^{3}d\sqrt{bd}+15\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}+60\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}cb{d}^{3}-270\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{c}^{2}{b}^{2}{d}^{2}+300\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{3}a{b}^{3}d-105\,{b}^{4}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{4}-30\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{a}^{3}{d}^{3}\sqrt{bd}+382\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{a}^{2}cb{d}^{2}\sqrt{bd}-530\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}a{c}^{2}{b}^{2}d\sqrt{bd}+210\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{c}^{3}{b}^{3}\sqrt{bd} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)

[Out]

-1/768*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(-96*x^6*b^3*d^3*(b*d*x^4+a*d*x^2+b*c*x^2
+a*c)^(1/2)*(b*d)^(1/2)-272*x^4*a*b^2*d^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b
*d)^(1/2)+112*x^4*b^3*c*d^2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)-236*
(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*a^2*b*d^3*(b*d)^(1/2)+344*(b*d*x^4+a*d*x
^2+b*c*x^2+a*c)^(1/2)*x^2*a*c*b^2*d^2*(b*d)^(1/2)-140*(b*d*x^4+a*d*x^2+b*c*x^2+a
*c)^(1/2)*x^2*c^2*b^3*d*(b*d)^(1/2)+15*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*
x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*d^4+60*ln(1/2*(2*b*d*x^2+2*
(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*c*b*d^
3-270*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*
c)/(b*d)^(1/2))*a^2*c^2*b^2*d^2+300*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2
+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c^3*a*b^3*d-105*b^4*ln(1/2*(2*b*d*
x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c^4-
30*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a^3*d^3*(b*d)^(1/2)+382*(b*d*x^4+a*d*x^2+
b*c*x^2+a*c)^(1/2)*a^2*c*b*d^2*(b*d)^(1/2)-530*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/
2)*a*c^2*b^2*d*(b*d)^(1/2)+210*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*c^3*b^3*(b*d)
^(1/2))/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d^4/(b*d)^(1/2)/b

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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((b*x^2 + a)^(5/2)*x^3/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296138, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{6} - 105 \, b^{3} c^{3} + 265 \, a b^{2} c^{2} d - 191 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 17 \, a b^{2} d^{3}\right )} x^{4} + 2 \,{\left (35 \, b^{3} c^{2} d - 86 \, a b^{2} c d^{2} + 59 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} - 15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{1536 \, \sqrt{b d} b d^{4}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{6} - 105 \, b^{3} c^{3} + 265 \, a b^{2} c^{2} d - 191 \, a^{2} b c d^{2} + 15 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 17 \, a b^{2} d^{3}\right )} x^{4} + 2 \,{\left (35 \, b^{3} c^{2} d - 86 \, a b^{2} c d^{2} + 59 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} + 15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{768 \, \sqrt{-b d} b d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1536*(4*(48*b^3*d^3*x^6 - 105*b^3*c^3 + 265*a*b^2*c^2*d - 191*a^2*b*c*d^2 + 1
5*a^3*d^3 - 8*(7*b^3*c*d^2 - 17*a*b^2*d^3)*x^4 + 2*(35*b^3*c^2*d - 86*a*b^2*c*d^
2 + 59*a^2*b*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b*d) - 15*(7*b^4*c^4
- 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*log(-4*(2*b^2*
d^2*x^2 + b^2*c*d + a*b*d^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c) + (8*b^2*d^2*x^4 +
b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x^2)*sqrt(b*d)))/(sqrt(b*d
)*b*d^4), 1/768*(2*(48*b^3*d^3*x^6 - 105*b^3*c^3 + 265*a*b^2*c^2*d - 191*a^2*b*c
*d^2 + 15*a^3*d^3 - 8*(7*b^3*c*d^2 - 17*a*b^2*d^3)*x^4 + 2*(35*b^3*c^2*d - 86*a*
b^2*c*d^2 + 59*a^2*b*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d) + 15*(
7*b^4*c^4 - 20*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*arcta
n(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*b*d)))
/(sqrt(-b*d)*b*d^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*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.255627, size = 397, normalized size = 1.68 \[ \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (2 \,{\left (b x^{2} + a\right )}{\left (4 \,{\left (b x^{2} + a\right )}{\left (\frac{6 \,{\left (b x^{2} + a\right )}}{b d} - \frac{7 \, b^{2} c d^{5} + a b d^{6}}{b^{2} d^{7}}\right )} + \frac{5 \,{\left (7 \, b^{3} c^{2} d^{4} - 6 \, a b^{2} c d^{5} - a^{2} b d^{6}\right )}}{b^{2} d^{7}}\right )} - \frac{15 \,{\left (7 \, b^{4} c^{3} d^{3} - 13 \, a b^{3} c^{2} d^{4} + 5 \, a^{2} b^{2} c d^{5} + a^{3} b d^{6}\right )}}{b^{2} d^{7}}\right )} - \frac{15 \,{\left (7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{4}}}{384 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)*(2*(b*x^2 + a)*(4*(
b*x^2 + a)*(6*(b*x^2 + a)/(b*d) - (7*b^2*c*d^5 + a*b*d^6)/(b^2*d^7)) + 5*(7*b^3*
c^2*d^4 - 6*a*b^2*c*d^5 - a^2*b*d^6)/(b^2*d^7)) - 15*(7*b^4*c^3*d^3 - 13*a*b^3*c
^2*d^4 + 5*a^2*b^2*c*d^5 + a^3*b*d^6)/(b^2*d^7)) - 15*(7*b^4*c^4 - 20*a*b^3*c^3*
d + 18*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - a^4*d^4)*ln(abs(-sqrt(b*x^2 + a)*sqrt(b
*d) + sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)))/(sqrt(b*d)*d^4))/abs(b)