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

Optimal. Leaf size=187 \[ -\frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{16 b^{3/2} d^{7/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d) (a d+5 b c)}{16 b d^3}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (a d+5 b c)}{24 b d^2}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 b d} \]

[Out]

((b*c - a*d)*(5*b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(16*b*d^3) - ((5*b*c
+ a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(24*b*d^2) + ((a + b*x^2)^(5/2)*Sqrt[
c + d*x^2])/(6*b*d) - ((b*c - a*d)^2*(5*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x
^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(16*b^(3/2)*d^(7/2))

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Rubi [A]  time = 0.429235, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{(b c-a d)^2 (a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{16 b^{3/2} d^{7/2}}+\frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d) (a d+5 b c)}{16 b d^3}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (a d+5 b c)}{24 b d^2}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2}}{6 b d} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*(5*b*c + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(16*b*d^3) - ((5*b*c
+ a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(24*b*d^2) + ((a + b*x^2)^(5/2)*Sqrt[
c + d*x^2])/(6*b*d) - ((b*c - a*d)^2*(5*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x
^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(16*b^(3/2)*d^(7/2))

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Rubi in Sympy [A]  time = 39.467, size = 163, normalized size = 0.87 \[ \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \sqrt{c + d x^{2}}}{6 b d} - \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d + 5 b c\right )}{24 b d^{2}} - \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (a d + 5 b c\right )}{16 b d^{3}} - \frac{\left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{16 b^{\frac{3}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**2)**(5/2)*sqrt(c + d*x**2)/(6*b*d) - (a + b*x**2)**(3/2)*sqrt(c + d*x*
*2)*(a*d + 5*b*c)/(24*b*d**2) - sqrt(a + b*x**2)*sqrt(c + d*x**2)*(a*d - b*c)*(a
*d + 5*b*c)/(16*b*d**3) - (a*d - b*c)**2*(a*d + 5*b*c)*atanh(sqrt(d)*sqrt(a + b*
x**2)/(sqrt(b)*sqrt(c + d*x**2)))/(16*b**(3/2)*d**(7/2))

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Mathematica [A]  time = 0.153012, size = 163, normalized size = 0.87 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (3 a^2 d^2+2 a b d \left (7 d x^2-11 c\right )+b^2 \left (15 c^2-10 c d x^2+8 d^2 x^4\right )\right )}{48 b d^3}-\frac{(b c-a d)^2 (a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{32 b^{3/2} d^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(3*a^2*d^2 + 2*a*b*d*(-11*c + 7*d*x^2) + b^2*(1
5*c^2 - 10*c*d*x^2 + 8*d^2*x^4)))/(48*b*d^3) - ((b*c - a*d)^2*(5*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]])/(32*
b^(3/2)*d^(7/2))

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Maple [B]  time = 0.024, size = 532, normalized size = 2.8 \[ -{\frac{1}{96\,{d}^{3}b}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( -16\,{x}^{4}{b}^{2}{d}^{2}\sqrt{bd}\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}-28\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}ab{d}^{2}\sqrt{bd}+20\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{x}^{2}c{b}^{2}d\sqrt{bd}+3\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{3}{d}^{3}+9\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){a}^{2}cb{d}^{2}-27\,\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{2}a{b}^{2}d+15\,{b}^{3}\ln \left ( 1/2\,{\frac{2\,bd{x}^{2}+2\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){c}^{3}-6\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{a}^{2}{d}^{2}\sqrt{bd}+44\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}acbd\sqrt{bd}-30\,\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}{c}^{2}{b}^{2}\sqrt{bd} \right ){\frac{1}{\sqrt{bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(-16*x^4*b^2*d^2*(b*d)^(1/2)*(b*d*x^4+a*d*
x^2+b*c*x^2+a*c)^(1/2)-28*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*a*b*d^2*(b*d)^
(1/2)+20*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*c*b^2*d*(b*d)^(1/2)+3*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*d^3+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^2*c*b*d^2-27*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^2*a*b^2*d+15*b^3*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^3-6
*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a^2*d^2*(b*d)^(1/2)+44*(b*d*x^4+a*d*x^2+b*c
*x^2+a*c)^(1/2)*a*c*b*d*(b*d)^(1/2)-30*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*c^2*b
^2*(b*d)^(1/2))/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d^3/b/(b*d)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273407, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, b^{2} d^{2} x^{4} + 15 \, b^{2} c^{2} - 22 \, a b c d + 3 \, a^{2} d^{2} - 2 \,{\left (5 \, b^{2} c d - 7 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} + 3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{192 \, \sqrt{b d} b d^{3}}, \frac{2 \,{\left (8 \, b^{2} d^{2} x^{4} + 15 \, b^{2} c^{2} - 22 \, a b c d + 3 \, a^{2} d^{2} - 2 \,{\left (5 \, b^{2} c d - 7 \, a b d^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} - 3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{96 \, \sqrt{-b d} b d^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/192*(4*(8*b^2*d^2*x^4 + 15*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2 - 2*(5*b^2*c*d -
7*a*b*d^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b*d) + 3*(5*b^3*c^3 - 9*a*b
^2*c^2*d + 3*a^2*b*c*d^2 + a^3*d^3)*log(-4*(2*b^2*d^2*x^2 + b^2*c*d + a*b*d^2)*s
qrt(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^3), 1/96*(2*(8*b^2*d^2*x
^4 + 15*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2 - 2*(5*b^2*c*d - 7*a*b*d^2)*x^2)*sqrt(b
*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d) - 3*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*
d^2 + a^3*d^3)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x^2 + a)*sq
rt(d*x^2 + c)*b*d)))/(sqrt(-b*d)*b*d^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.260105, size = 293, normalized size = 1.57 \[ \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (2 \,{\left (b x^{2} + a\right )}{\left (\frac{4 \,{\left (b x^{2} + a\right )}}{b d} - \frac{5 \, b^{2} c d^{3} + a b d^{4}}{b^{2} d^{5}}\right )} + \frac{3 \,{\left (5 \, b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} - a^{2} b d^{4}\right )}}{b^{2} d^{5}}\right )} + \frac{3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{3}}}{48 \,{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(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)/(b*d) - (5*b^2*c*d^3 + a*b*d^4)/(b^2*d^5)) + 3*(5*b^3*c^2*d^2 - 4*a*b^
2*c*d^3 - a^2*b*d^4)/(b^2*d^5)) + 3*(5*b^3*c^3 - 9*a*b^2*c^2*d + 3*a^2*b*c*d^2 +
a^3*d^3)*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^3))/abs(b)