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

Optimal. Leaf size=171 \[ \frac{315 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{32 a^{11/2}}-\frac{315 b^3 \sqrt{a x+b \sqrt{x}}}{32 a^5}+\frac{105 b^2 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{16 a^4}-\frac{21 b x \sqrt{a x+b \sqrt{x}}}{4 a^3}+\frac{9 x^{3/2} \sqrt{a x+b \sqrt{x}}}{2 a^2}-\frac{4 x^{5/2}}{a \sqrt{a x+b \sqrt{x}}} \]

[Out]

(-4*x^(5/2))/(a*Sqrt[b*Sqrt[x] + a*x]) - (315*b^3*Sqrt[b*Sqrt[x] + a*x])/(32*a^5
) + (105*b^2*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/(16*a^4) - (21*b*x*Sqrt[b*Sqrt[x] +
a*x])/(4*a^3) + (9*x^(3/2)*Sqrt[b*Sqrt[x] + a*x])/(2*a^2) + (315*b^4*ArcTanh[(Sq
rt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(32*a^(11/2))

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Rubi [A]  time = 0.319789, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{315 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{32 a^{11/2}}-\frac{315 b^3 \sqrt{a x+b \sqrt{x}}}{32 a^5}+\frac{105 b^2 \sqrt{x} \sqrt{a x+b \sqrt{x}}}{16 a^4}-\frac{21 b x \sqrt{a x+b \sqrt{x}}}{4 a^3}+\frac{9 x^{3/2} \sqrt{a x+b \sqrt{x}}}{2 a^2}-\frac{4 x^{5/2}}{a \sqrt{a x+b \sqrt{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*x^(5/2))/(a*Sqrt[b*Sqrt[x] + a*x]) - (315*b^3*Sqrt[b*Sqrt[x] + a*x])/(32*a^5
) + (105*b^2*Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/(16*a^4) - (21*b*x*Sqrt[b*Sqrt[x] +
a*x])/(4*a^3) + (9*x^(3/2)*Sqrt[b*Sqrt[x] + a*x])/(2*a^2) + (315*b^4*ArcTanh[(Sq
rt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(32*a^(11/2))

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Rubi in Sympy [A]  time = 33.5602, size = 160, normalized size = 0.94 \[ - \frac{4 x^{\frac{5}{2}}}{a \sqrt{a x + b \sqrt{x}}} + \frac{9 x^{\frac{3}{2}} \sqrt{a x + b \sqrt{x}}}{2 a^{2}} - \frac{21 b x \sqrt{a x + b \sqrt{x}}}{4 a^{3}} + \frac{105 b^{2} \sqrt{x} \sqrt{a x + b \sqrt{x}}}{16 a^{4}} - \frac{315 b^{3} \sqrt{a x + b \sqrt{x}}}{32 a^{5}} + \frac{315 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{32 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*x**(5/2)/(a*sqrt(a*x + b*sqrt(x))) + 9*x**(3/2)*sqrt(a*x + b*sqrt(x))/(2*a**2
) - 21*b*x*sqrt(a*x + b*sqrt(x))/(4*a**3) + 105*b**2*sqrt(x)*sqrt(a*x + b*sqrt(x
))/(16*a**4) - 315*b**3*sqrt(a*x + b*sqrt(x))/(32*a**5) + 315*b**4*atanh(sqrt(a)
*sqrt(x)/sqrt(a*x + b*sqrt(x)))/(32*a**(11/2))

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Mathematica [A]  time = 0.189856, size = 124, normalized size = 0.73 \[ \frac{315 b^4 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{64 a^{11/2}}+\frac{\sqrt{a x+b \sqrt{x}} \left (16 a^4 x^2-24 a^3 b x^{3/2}+42 a^2 b^2 x-105 a b^3 \sqrt{x}-315 b^4\right )}{32 a^5 \left (a \sqrt{x}+b\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b*Sqrt[x] + a*x]*(-315*b^4 - 105*a*b^3*Sqrt[x] + 42*a^2*b^2*x - 24*a^3*b*x
^(3/2) + 16*a^4*x^2))/(32*a^5*(b + a*Sqrt[x])) + (315*b^4*Log[b + 2*a*Sqrt[x] +
2*Sqrt[a]*Sqrt[b*Sqrt[x] + a*x]])/(64*a^(11/2))

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Maple [B]  time = 0.014, size = 531, normalized size = 3.1 \[{\frac{1}{64}\sqrt{b\sqrt{x}+ax} \left ( 32\,{x}^{3/2} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{19/2}+276\,{x}^{3/2}\sqrt{b\sqrt{x}+ax}{a}^{17/2}{b}^{2}-48\,x \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{17/2}b-768\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }x{a}^{15/2}{b}^{3}+690\,x\sqrt{b\sqrt{x}+ax}{a}^{15/2}{b}^{3}-192\,\sqrt{x} \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{15/2}{b}^{2}+256\,{b}^{3}{a}^{13/2} \left ( \sqrt{x} \left ( b+\sqrt{x}a \right ) \right ) ^{3/2}-1536\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{x}{a}^{13/2}{b}^{4}+552\,\sqrt{x}\sqrt{b\sqrt{x}+ax}{a}^{13/2}{b}^{4}-112\, \left ( b\sqrt{x}+ax \right ) ^{3/2}{a}^{13/2}{b}^{3}-768\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{a}^{11/2}{b}^{5}+138\,\sqrt{b\sqrt{x}+ax}{a}^{11/2}{b}^{5}-69\,x\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7}{b}^{4}+384\,x\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{7}{b}^{4}-138\,\sqrt{x}\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{6}{b}^{5}+768\,\sqrt{x}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{6}{b}^{5}-69\,\ln \left ( 1/2\,{\frac{2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5}{b}^{6}+384\,\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{5}{b}^{6} \right ){a}^{-{\frac{21}{2}}}{\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}} \left ( b+\sqrt{x}a \right ) ^{-2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64*(b*x^(1/2)+a*x)^(1/2)/a^(21/2)*(32*x^(3/2)*(b*x^(1/2)+a*x)^(3/2)*a^(19/2)+2
76*x^(3/2)*(b*x^(1/2)+a*x)^(1/2)*a^(17/2)*b^2-48*x*(b*x^(1/2)+a*x)^(3/2)*a^(17/2
)*b-768*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x*a^(15/2)*b^3+690*x*(b*x^(1/2)+a*x)^(1/2)
*a^(15/2)*b^3-192*x^(1/2)*(b*x^(1/2)+a*x)^(3/2)*a^(15/2)*b^2+256*b^3*a^(13/2)*(x
^(1/2)*(b+x^(1/2)*a))^(3/2)-1536*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*x^(1/2)*a^(13/2)*
b^4+552*x^(1/2)*(b*x^(1/2)+a*x)^(1/2)*a^(13/2)*b^4-112*(b*x^(1/2)+a*x)^(3/2)*a^(
13/2)*b^3-768*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*a^(11/2)*b^5+138*(b*x^(1/2)+a*x)^(1/
2)*a^(11/2)*b^5-69*x*ln(1/2*(2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+2*x^(1/2)*a+b)/a^(1
/2))*a^7*b^4+384*x*ln(1/2*(2*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*a^(1/2)+2*x^(1/2)*a+b
)/a^(1/2))*a^7*b^4-138*x^(1/2)*ln(1/2*(2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+2*x^(1/2)
*a+b)/a^(1/2))*a^6*b^5+768*x^(1/2)*ln(1/2*(2*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*a^(1/
2)+2*x^(1/2)*a+b)/a^(1/2))*a^6*b^5-69*ln(1/2*(2*(b*x^(1/2)+a*x)^(1/2)*a^(1/2)+2*
x^(1/2)*a+b)/a^(1/2))*a^5*b^6+384*ln(1/2*(2*(x^(1/2)*(b+x^(1/2)*a))^(1/2)*a^(1/2
)+2*x^(1/2)*a+b)/a^(1/2))*a^5*b^6)/(x^(1/2)*(b+x^(1/2)*a))^(1/2)/(b+x^(1/2)*a)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(5/2)/(a*x + b*sqrt(x))^(3/2), x)

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError