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

Optimal. Leaf size=276 \[ -\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{128 b^2 d^4}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{192 b^2 d^3}+\frac{(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{128 b^{5/2} d^{9/2}}-\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2} (3 a d+7 b c)}{48 b^2 d^2}+\frac{x^2 \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{8 b d} \]

[Out]

-((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^2]*Sqrt[c + d*x
^2])/(128*b^2*d^4) + ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)^(3/2)*Sq
rt[c + d*x^2])/(192*b^2*d^3) - ((7*b*c + 3*a*d)*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2
])/(48*b^2*d^2) + (x^2*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(8*b*d) + ((b*c - a*d)
^2*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt
[b]*Sqrt[c + d*x^2])])/(128*b^(5/2)*d^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.763458, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d) \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{128 b^2 d^4}+\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right )}{192 b^2 d^3}+\frac{(b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{128 b^{5/2} d^{9/2}}-\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2} (3 a d+7 b c)}{48 b^2 d^2}+\frac{x^2 \left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{8 b d} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^2]*Sqrt[c + d*x
^2])/(128*b^2*d^4) + ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)^(3/2)*Sq
rt[c + d*x^2])/(192*b^2*d^3) - ((7*b*c + 3*a*d)*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2
])/(48*b^2*d^2) + (x^2*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(8*b*d) + ((b*c - a*d)
^2*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x^2])/(Sqrt
[b]*Sqrt[c + d*x^2])])/(128*b^(5/2)*d^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 61.6349, size = 260, normalized size = 0.94 \[ \frac{x^{2} \left (a + b x^{2}\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}}{8 b d} - \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \sqrt{c + d x^{2}} \left (3 a d + 7 b c\right )}{48 b^{2} d^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right )}{192 b^{2} d^{3}} + \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right )}{128 b^{2} d^{4}} + \frac{\left (a d - b c\right )^{2} \left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{128 b^{\frac{5}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(a + b*x**2)**(5/2)*sqrt(c + d*x**2)/(8*b*d) - (a + b*x**2)**(5/2)*sqrt(c +
d*x**2)*(3*a*d + 7*b*c)/(48*b**2*d**2) + (a + b*x**2)**(3/2)*sqrt(c + d*x**2)*(
3*a**2*d**2 + 10*a*b*c*d + 35*b**2*c**2)/(192*b**2*d**3) + sqrt(a + b*x**2)*sqrt
(c + d*x**2)*(a*d - b*c)*(3*a**2*d**2 + 10*a*b*c*d + 35*b**2*c**2)/(128*b**2*d**
4) + (a*d - b*c)**2*(3*a**2*d**2 + 10*a*b*c*d + 35*b**2*c**2)*atanh(sqrt(d)*sqrt
(a + b*x**2)/(sqrt(b)*sqrt(c + d*x**2)))/(128*b**(5/2)*d**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.261225, size = 224, normalized size = 0.81 \[ \frac{3 (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )-2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2} \left (9 a^3 d^3+3 a^2 b d^2 \left (5 c-2 d x^2\right )+a b^2 d \left (-145 c^2+92 c d x^2-72 d^2 x^4\right )+b^3 \left (105 c^3-70 c^2 d x^2+56 c d^2 x^4-48 d^3 x^6\right )\right )}{768 b^{5/2} d^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(9*a^3*d^3 + 3*a^2*b*d^2*(5*
c - 2*d*x^2) + a*b^2*d*(-145*c^2 + 92*c*d*x^2 - 72*d^2*x^4) + b^3*(105*c^3 - 70*
c^2*d*x^2 + 56*c*d^2*x^4 - 48*d^3*x^6)) + 3*(b*c - a*d)^2*(35*b^2*c^2 + 10*a*b*c
*d + 3*a^2*d^2)*Log[b*c + a*d + 2*b*d*x^2 + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x^2]*Sq
rt[c + d*x^2]])/(768*b^(5/2)*d^(9/2))

_______________________________________________________________________________________

Maple [B]  time = 0.046, size = 770, normalized size = 2.8 \[{\frac{1}{768\,{b}^{2}{d}^{4}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 96\,{x}^{6}{b}^{3}{d}^{3}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+144\,{x}^{4}a{b}^{2}{d}^{3}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}-112\,{x}^{4}{b}^{3}c{d}^{2}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+12\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}{a}^{2}b{d}^{3}\sqrt{bd}-184\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}ac{b}^{2}{d}^{2}\sqrt{bd}+140\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}{c}^{2}{b}^{3}d\sqrt{bd}+9\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{4}{d}^{4}+12\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}cb{d}^{3}+54\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}{c}^{2}{b}^{2}{d}^{2}-180\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{3}a{b}^{3}d+105\,{b}^{4}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{4}-18\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{a}^{3}{d}^{3}\sqrt{bd}-30\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{a}^{2}cb{d}^{2}\sqrt{bd}+290\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}a{c}^{2}{b}^{2}d\sqrt{bd}-210\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{c}^{3}{b}^{3}\sqrt{bd} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^(3/2)/(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)+144*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)+12*(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)-184*(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)+9*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+12*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+54
*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^2-180*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^4-18*(b
*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a^3*d^3*(b*d)^(1/2)-30*(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)+290*(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)/b^2/d^4/(b*d)^(1/2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.300149, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, b^{3} d^{3} x^{6} - 105 \, b^{3} c^{3} + 145 \, a b^{2} c^{2} d - 15 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 9 \, a b^{2} d^{3}\right )} x^{4} + 2 \,{\left (35 \, b^{3} c^{2} d - 46 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} + 3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{1536 \, \sqrt{b d} b^{2} d^{4}}, \frac{2 \,{\left (48 \, b^{3} d^{3} x^{6} - 105 \, b^{3} c^{3} + 145 \, a b^{2} c^{2} d - 15 \, a^{2} b c d^{2} - 9 \, a^{3} d^{3} - 8 \,{\left (7 \, b^{3} c d^{2} - 9 \, a b^{2} d^{3}\right )} x^{4} + 2 \,{\left (35 \, b^{3} c^{2} d - 46 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} + 3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{768 \, \sqrt{-b d} b^{2} d^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1536*(4*(48*b^3*d^3*x^6 - 105*b^3*c^3 + 145*a*b^2*c^2*d - 15*a^2*b*c*d^2 - 9*
a^3*d^3 - 8*(7*b^3*c*d^2 - 9*a*b^2*d^3)*x^4 + 2*(35*b^3*c^2*d - 46*a*b^2*c*d^2 +
3*a^2*b*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b*d) + 3*(35*b^4*c^4 - 6
0*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 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
^2*d^4), 1/768*(2*(48*b^3*d^3*x^6 - 105*b^3*c^3 + 145*a*b^2*c^2*d - 15*a^2*b*c*d
^2 - 9*a^3*d^3 - 8*(7*b^3*c*d^2 - 9*a*b^2*d^3)*x^4 + 2*(35*b^3*c^2*d - 46*a*b^2*
c*d^2 + 3*a^2*b*d^3)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d) + 3*(35*b^4
*c^4 - 60*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 3*a^4*d^4)*arctan(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)))/(s
qrt(-b*d)*b^2*d^4)]

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.25367, size = 410, normalized size = 1.49 \[ \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (2 \,{\left (b x^{2} + a\right )}{\left (4 \,{\left (b x^{2} + a\right )}{\left (\frac{6 \,{\left (b x^{2} + a\right )}}{b d} - \frac{7 \, b^{3} c d^{5} + 9 \, a b^{2} d^{6}}{b^{3} d^{7}}\right )} + \frac{35 \, b^{4} c^{2} d^{4} + 10 \, a b^{3} c d^{5} + 3 \, a^{2} b^{2} d^{6}}{b^{3} d^{7}}\right )} - \frac{3 \,{\left (35 \, b^{5} c^{3} d^{3} - 25 \, a b^{4} c^{2} d^{4} - 7 \, a^{2} b^{3} c d^{5} - 3 \, a^{3} b^{2} d^{6}\right )}}{b^{3} d^{7}}\right )} - \frac{3 \,{\left (35 \, b^{4} c^{4} - 60 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 3 \, a^{4} d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{4}}}{384 \, b{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(3/2)*x^5/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^3*c*d^5 + 9*a*b^2*d^6)/(b^3*d^7)) + (35*b
^4*c^2*d^4 + 10*a*b^3*c*d^5 + 3*a^2*b^2*d^6)/(b^3*d^7)) - 3*(35*b^5*c^3*d^3 - 25
*a*b^4*c^2*d^4 - 7*a^2*b^3*c*d^5 - 3*a^3*b^2*d^6)/(b^3*d^7)) - 3*(35*b^4*c^4 - 6
0*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 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))/(b*
abs(b))