3.359 \(\int \frac{x^5 \sqrt{c+d x^3}}{a+b x^3} \, dx\)

Optimal. Leaf size=93 \[ \frac{2 a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2}}-\frac{2 a \sqrt{c+d x^3}}{3 b^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d} \]

[Out]

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

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Rubi [A]  time = 0.210128, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{2 a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2}}-\frac{2 a \sqrt{c+d x^3}}{3 b^2}+\frac{2 \left (c+d x^3\right )^{3/2}}{9 b d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.8656, size = 82, normalized size = 0.88 \[ - \frac{2 a \sqrt{c + d x^{3}}}{3 b^{2}} + \frac{2 a \sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{3}}}{\sqrt{a d - b c}} \right )}}{3 b^{\frac{5}{2}}} + \frac{2 \left (c + d x^{3}\right )^{\frac{3}{2}}}{9 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.299215, size = 88, normalized size = 0.95 \[ \frac{2 a \sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 b^{5/2}}+\frac{2 \sqrt{c+d x^3} \left (b \left (c+d x^3\right )-3 a d\right )}{9 b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c + d*x^3]*(-3*a*d + b*(c + d*x^3)))/(9*b^2*d) + (2*a*Sqrt[b*c - a*d]*Ar
cTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(3*b^(5/2))

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Maple [C]  time = 0.012, size = 458, normalized size = 4.9 \[{\frac{2}{9\,bd} \left ( d{x}^{3}+c \right ) ^{{\frac{3}{2}}}}-{\frac{a}{b} \left ({\frac{2}{3\,b}\sqrt{d{x}^{3}+c}}+{\frac{{\frac{i}{3}}\sqrt{2}}{b{d}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{1\sqrt [3]{-c{d}^{2}}\sqrt{{{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( -i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}\sqrt{{d \left ( x-{\frac{1}{d}\sqrt [3]{-c{d}^{2}}} \right ) \left ( -3\,\sqrt [3]{-c{d}^{2}}+i\sqrt{3}\sqrt [3]{-c{d}^{2}} \right ) ^{-1}}}\sqrt{{-{\frac{i}{2}}d \left ( 2\,x+{\frac{1}{d} \left ( i\sqrt{3}\sqrt [3]{-c{d}^{2}}+\sqrt [3]{-c{d}^{2}} \right ) } \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}} \left ( i\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,\sqrt{3}d+2\,{{\it \_alpha}}^{2}{d}^{2}-i\sqrt{3} \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}-\sqrt [3]{-c{d}^{2}}{\it \_alpha}\,d- \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}} \right ){\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{{i\sqrt{3}d \left ( x+{\frac{1}{2\,d}\sqrt [3]{-c{d}^{2}}}-{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ){\frac{1}{\sqrt [3]{-c{d}^{2}}}}}}},{\frac{b}{2\, \left ( ad-bc \right ) d} \left ( 2\,i{{\it \_alpha}}^{2}\sqrt [3]{-c{d}^{2}}\sqrt{3}d-i{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{{\frac{2}{3}}}\sqrt{3}+i\sqrt{3}cd-3\,{\it \_alpha}\, \left ( -c{d}^{2} \right ) ^{2/3}-3\,cd \right ) },\sqrt{{\frac{i\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}} \left ( -{\frac{3}{2\,d}\sqrt [3]{-c{d}^{2}}}+{\frac{{\frac{i}{2}}\sqrt{3}}{d}\sqrt [3]{-c{d}^{2}}} \right ) ^{-1}}} \right ){\frac{1}{\sqrt{d{x}^{3}+c}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(d*x^3+c)^(3/2)/b/d-a/b*(2/3*(d*x^3+c)^(1/2)/b+1/3*I/b/d^2*2^(1/2)*sum((-c*d
^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)
^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1
/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^
2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*d^
2-I*3^(1/2)*(-c*d^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/
3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(
-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c
*d^2)^(2/3)*3^(1/2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/(a*d-b*c),(I*3^
(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(
1/2)),_alpha=RootOf(_Z^3*b+a)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222713, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a d \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \,{\left (b d x^{3} + b c - 3 \, a d\right )} \sqrt{d x^{3} + c}}{9 \, b^{2} d}, \frac{2 \,{\left (3 \, a d \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-\frac{b c - a d}{b}}}\right ) +{\left (b d x^{3} + b c - 3 \, a d\right )} \sqrt{d x^{3} + c}\right )}}{9 \, b^{2} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/9*(3*a*d*sqrt((b*c - a*d)/b)*log((b*d*x^3 + 2*b*c - a*d + 2*sqrt(d*x^3 + c)*b
*sqrt((b*c - a*d)/b))/(b*x^3 + a)) + 2*(b*d*x^3 + b*c - 3*a*d)*sqrt(d*x^3 + c))/
(b^2*d), 2/9*(3*a*d*sqrt(-(b*c - a*d)/b)*arctan(sqrt(d*x^3 + c)/sqrt(-(b*c - a*d
)/b)) + (b*d*x^3 + b*c - 3*a*d)*sqrt(d*x^3 + c))/(b^2*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216413, size = 130, normalized size = 1.4 \[ -\frac{2 \,{\left (\frac{3 \,{\left (a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{2}} - \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}} b^{2} - 3 \, \sqrt{d x^{3} + c} a b d}{b^{3}}\right )}}{9 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(3*(a*b*c*d - a^2*d^2)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt
(-b^2*c + a*b*d)*b^2) - ((d*x^3 + c)^(3/2)*b^2 - 3*sqrt(d*x^3 + c)*a*b*d)/b^3)/d