3.182 \(\int \frac{\left (b x^{2/3}+a x\right )^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=203 \[ -\frac{21 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{512 b^{9/2}}+\frac{21 a^5 \sqrt{a x+b x^{2/3}}}{512 b^4 x^{2/3}}-\frac{7 a^4 \sqrt{a x+b x^{2/3}}}{256 b^3 x}+\frac{7 a^3 \sqrt{a x+b x^{2/3}}}{320 b^2 x^{4/3}}-\frac{3 a^2 \sqrt{a x+b x^{2/3}}}{160 b x^{5/3}}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{2 x^3}-\frac{3 a \sqrt{a x+b x^{2/3}}}{20 x^2} \]

[Out]

(-3*a*Sqrt[b*x^(2/3) + a*x])/(20*x^2) - (3*a^2*Sqrt[b*x^(2/3) + a*x])/(160*b*x^(
5/3)) + (7*a^3*Sqrt[b*x^(2/3) + a*x])/(320*b^2*x^(4/3)) - (7*a^4*Sqrt[b*x^(2/3)
+ a*x])/(256*b^3*x) + (21*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^4*x^(2/3)) - (b*x^(2
/3) + a*x)^(3/2)/(2*x^3) - (21*a^6*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*
x]])/(512*b^(9/2))

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Rubi [A]  time = 0.582914, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{21 a^6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{512 b^{9/2}}+\frac{21 a^5 \sqrt{a x+b x^{2/3}}}{512 b^4 x^{2/3}}-\frac{7 a^4 \sqrt{a x+b x^{2/3}}}{256 b^3 x}+\frac{7 a^3 \sqrt{a x+b x^{2/3}}}{320 b^2 x^{4/3}}-\frac{3 a^2 \sqrt{a x+b x^{2/3}}}{160 b x^{5/3}}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{2 x^3}-\frac{3 a \sqrt{a x+b x^{2/3}}}{20 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^(2/3) + a*x)^(3/2)/x^4,x]

[Out]

(-3*a*Sqrt[b*x^(2/3) + a*x])/(20*x^2) - (3*a^2*Sqrt[b*x^(2/3) + a*x])/(160*b*x^(
5/3)) + (7*a^3*Sqrt[b*x^(2/3) + a*x])/(320*b^2*x^(4/3)) - (7*a^4*Sqrt[b*x^(2/3)
+ a*x])/(256*b^3*x) + (21*a^5*Sqrt[b*x^(2/3) + a*x])/(512*b^4*x^(2/3)) - (b*x^(2
/3) + a*x)^(3/2)/(2*x^3) - (21*a^6*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*
x]])/(512*b^(9/2))

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Rubi in Sympy [A]  time = 50.8292, size = 187, normalized size = 0.92 \[ - \frac{21 a^{6} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x + b x^{\frac{2}{3}}}} \right )}}{512 b^{\frac{9}{2}}} + \frac{21 a^{5} \sqrt{a x + b x^{\frac{2}{3}}}}{512 b^{4} x^{\frac{2}{3}}} - \frac{7 a^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{256 b^{3} x} + \frac{7 a^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{320 b^{2} x^{\frac{4}{3}}} - \frac{3 a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{160 b x^{\frac{5}{3}}} - \frac{3 a \sqrt{a x + b x^{\frac{2}{3}}}}{20 x^{2}} - \frac{\left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{2 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**(2/3)+a*x)**(3/2)/x**4,x)

[Out]

-21*a**6*atanh(sqrt(b)*x**(1/3)/sqrt(a*x + b*x**(2/3)))/(512*b**(9/2)) + 21*a**5
*sqrt(a*x + b*x**(2/3))/(512*b**4*x**(2/3)) - 7*a**4*sqrt(a*x + b*x**(2/3))/(256
*b**3*x) + 7*a**3*sqrt(a*x + b*x**(2/3))/(320*b**2*x**(4/3)) - 3*a**2*sqrt(a*x +
b*x**(2/3))/(160*b*x**(5/3)) - 3*a*sqrt(a*x + b*x**(2/3))/(20*x**2) - (a*x + b*
x**(2/3))**(3/2)/(2*x**3)

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Mathematica [A]  time = 0.237915, size = 127, normalized size = 0.63 \[ \frac{\sqrt{a x+b x^{2/3}} \left (105 a^5 x^{5/3}-70 a^4 b x^{4/3}+56 a^3 b^2 x-48 a^2 b^3 x^{2/3}-1664 a b^4 \sqrt [3]{x}-1280 b^5\right )}{2560 b^4 x^{7/3}}-\frac{21 a^6 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{512 b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^(2/3) + a*x)^(3/2)/x^4,x]

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-1280*b^5 - 1664*a*b^4*x^(1/3) - 48*a^2*b^3*x^(2/3) + 56
*a^3*b^2*x - 70*a^4*b*x^(4/3) + 105*a^5*x^(5/3)))/(2560*b^4*x^(7/3)) - (21*a^6*A
rcTanh[Sqrt[b*x^(2/3) + a*x]/(Sqrt[b]*x^(1/3))])/(512*b^(9/2))

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Maple [A]  time = 0.019, size = 139, normalized size = 0.7 \[{\frac{1}{2560\,{x}^{3}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( 105\, \left ( b+a\sqrt [3]{x} \right ) ^{11/2}{b}^{9/2}-595\, \left ( b+a\sqrt [3]{x} \right ) ^{9/2}{b}^{11/2}+1386\, \left ( b+a\sqrt [3]{x} \right ) ^{7/2}{b}^{13/2}-1686\, \left ( b+a\sqrt [3]{x} \right ) ^{5/2}{b}^{15/2}-595\, \left ( b+a\sqrt [3]{x} \right ) ^{3/2}{b}^{17/2}+105\,\sqrt{b+a\sqrt [3]{x}}{b}^{19/2}-105\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){a}^{6}{b}^{4}{x}^{2} \right ) \left ( b+a\sqrt [3]{x} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{17}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^(2/3)+a*x)^(3/2)/x^4,x)

[Out]

1/2560*(b*x^(2/3)+a*x)^(3/2)*(105*(b+a*x^(1/3))^(11/2)*b^(9/2)-595*(b+a*x^(1/3))
^(9/2)*b^(11/2)+1386*(b+a*x^(1/3))^(7/2)*b^(13/2)-1686*(b+a*x^(1/3))^(5/2)*b^(15
/2)-595*(b+a*x^(1/3))^(3/2)*b^(17/2)+105*(b+a*x^(1/3))^(1/2)*b^(19/2)-105*arctan
h((b+a*x^(1/3))^(1/2)/b^(1/2))*a^6*b^4*x^2)/x^3/(b+a*x^(1/3))^(3/2)/b^(17/2)

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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((a*x + b*x^(2/3))^(3/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(2/3))^(3/2)/x^4,x, algorithm="fricas")

[Out]

Timed out

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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((b*x**(2/3)+a*x)**(3/2)/x**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.314681, size = 231, normalized size = 1.14 \[ \frac{\frac{105 \, a^{7} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right ){\rm sign}\left (x^{\frac{1}{3}}\right )}{\sqrt{-b} b^{4}} + \frac{105 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{7}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 595 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{7} b{\rm sign}\left (x^{\frac{1}{3}}\right ) + 1386 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{7} b^{2}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 1686 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{7} b^{3}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 595 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{7} b^{4}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 105 \, \sqrt{a x^{\frac{1}{3}} + b} a^{7} b^{5}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{6} b^{4} x^{2}}}{2560 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(2/3))^(3/2)/x^4,x, algorithm="giac")

[Out]

1/2560*(105*a^7*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))*sign(x^(1/3))/(sqrt(-b)*b^4
) + (105*(a*x^(1/3) + b)^(11/2)*a^7*sign(x^(1/3)) - 595*(a*x^(1/3) + b)^(9/2)*a^
7*b*sign(x^(1/3)) + 1386*(a*x^(1/3) + b)^(7/2)*a^7*b^2*sign(x^(1/3)) - 1686*(a*x
^(1/3) + b)^(5/2)*a^7*b^3*sign(x^(1/3)) - 595*(a*x^(1/3) + b)^(3/2)*a^7*b^4*sign
(x^(1/3)) + 105*sqrt(a*x^(1/3) + b)*a^7*b^5*sign(x^(1/3)))/(a^6*b^4*x^2))/a