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

Optimal. Leaf size=117 \[ \frac{a^4 c^2 \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a^3 c^2 \sqrt{c x^2}}{b^4}+\frac{a^2 c^2 x \sqrt{c x^2}}{2 b^3}-\frac{a c^2 x^2 \sqrt{c x^2}}{3 b^2}+\frac{c^2 x^3 \sqrt{c x^2}}{4 b} \]

[Out]

-((a^3*c^2*Sqrt[c*x^2])/b^4) + (a^2*c^2*x*Sqrt[c*x^2])/(2*b^3) - (a*c^2*x^2*Sqrt
[c*x^2])/(3*b^2) + (c^2*x^3*Sqrt[c*x^2])/(4*b) + (a^4*c^2*Sqrt[c*x^2]*Log[a + b*
x])/(b^5*x)

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Rubi [A]  time = 0.0817061, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a^4 c^2 \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a^3 c^2 \sqrt{c x^2}}{b^4}+\frac{a^2 c^2 x \sqrt{c x^2}}{2 b^3}-\frac{a c^2 x^2 \sqrt{c x^2}}{3 b^2}+\frac{c^2 x^3 \sqrt{c x^2}}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

-((a^3*c^2*Sqrt[c*x^2])/b^4) + (a^2*c^2*x*Sqrt[c*x^2])/(2*b^3) - (a*c^2*x^2*Sqrt
[c*x^2])/(3*b^2) + (c^2*x^3*Sqrt[c*x^2])/(4*b) + (a^4*c^2*Sqrt[c*x^2]*Log[a + b*
x])/(b^5*x)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{4} c^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{5} x} + \frac{a^{2} c^{2} \sqrt{c x^{2}} \int x\, dx}{b^{3} x} - \frac{a c^{2} x^{2} \sqrt{c x^{2}}}{3 b^{2}} + \frac{c^{2} x^{3} \sqrt{c x^{2}}}{4 b} - \frac{c^{2} \sqrt{c x^{2}} \int a^{3}\, dx}{b^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**4*c**2*sqrt(c*x**2)*log(a + b*x)/(b**5*x) + a**2*c**2*sqrt(c*x**2)*Integral(x
, x)/(b**3*x) - a*c**2*x**2*sqrt(c*x**2)/(3*b**2) + c**2*x**3*sqrt(c*x**2)/(4*b)
- c**2*sqrt(c*x**2)*Integral(a**3, x)/(b**4*x)

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Mathematica [A]  time = 0.0232445, size = 65, normalized size = 0.56 \[ \frac{c \left (c x^2\right )^{3/2} \left (12 a^4 \log (a+b x)+b x \left (-12 a^3+6 a^2 b x-4 a b^2 x^2+3 b^3 x^3\right )\right )}{12 b^5 x^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*(c*x^2)^(3/2)*(b*x*(-12*a^3 + 6*a^2*b*x - 4*a*b^2*x^2 + 3*b^3*x^3) + 12*a^4*L
og[a + b*x]))/(12*b^5*x^3)

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Maple [A]  time = 0.009, size = 63, normalized size = 0.5 \[{\frac{3\,{x}^{4}{b}^{4}-4\,{x}^{3}a{b}^{3}+6\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) -12\,x{a}^{3}b}{12\,{b}^{5}{x}^{5}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)/x/(b*x+a),x)

[Out]

1/12*(c*x^2)^(5/2)*(3*x^4*b^4-4*x^3*a*b^3+6*x^2*a^2*b^2+12*a^4*ln(b*x+a)-12*x*a^
3*b)/b^5/x^5

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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((c*x^2)^(5/2)/((b*x + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22976, size = 104, normalized size = 0.89 \[ \frac{{\left (3 \, b^{4} c^{2} x^{4} - 4 \, a b^{3} c^{2} x^{3} + 6 \, a^{2} b^{2} c^{2} x^{2} - 12 \, a^{3} b c^{2} x + 12 \, a^{4} c^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{12 \, b^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*b^4*c^2*x^4 - 4*a*b^3*c^2*x^3 + 6*a^2*b^2*c^2*x^2 - 12*a^3*b*c^2*x + 12*
a^4*c^2*log(b*x + a))*sqrt(c*x^2)/(b^5*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.212035, size = 134, normalized size = 1.15 \[ \frac{1}{12} \,{\left (\frac{12 \, a^{4} c^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{5}} - \frac{12 \, a^{4} c^{2}{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{5}} + \frac{3 \, b^{3} c^{2} x^{4}{\rm sign}\left (x\right ) - 4 \, a b^{2} c^{2} x^{3}{\rm sign}\left (x\right ) + 6 \, a^{2} b c^{2} x^{2}{\rm sign}\left (x\right ) - 12 \, a^{3} c^{2} x{\rm sign}\left (x\right )}{b^{4}}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(12*a^4*c^2*ln(abs(b*x + a))*sign(x)/b^5 - 12*a^4*c^2*ln(abs(a))*sign(x)/b^
5 + (3*b^3*c^2*x^4*sign(x) - 4*a*b^2*c^2*x^3*sign(x) + 6*a^2*b*c^2*x^2*sign(x) -
12*a^3*c^2*x*sign(x))/b^4)*sqrt(c)