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

Optimal. Leaf size=142 \[ -\frac{a^5 c^2 \sqrt{c x^2} \log (a+b x)}{b^6 x}+\frac{a^4 c^2 \sqrt{c x^2}}{b^5}-\frac{a^3 c^2 x \sqrt{c x^2}}{2 b^4}+\frac{a^2 c^2 x^2 \sqrt{c x^2}}{3 b^3}-\frac{a c^2 x^3 \sqrt{c x^2}}{4 b^2}+\frac{c^2 x^4 \sqrt{c x^2}}{5 b} \]

[Out]

(a^4*c^2*Sqrt[c*x^2])/b^5 - (a^3*c^2*x*Sqrt[c*x^2])/(2*b^4) + (a^2*c^2*x^2*Sqrt[
c*x^2])/(3*b^3) - (a*c^2*x^3*Sqrt[c*x^2])/(4*b^2) + (c^2*x^4*Sqrt[c*x^2])/(5*b)
- (a^5*c^2*Sqrt[c*x^2]*Log[a + b*x])/(b^6*x)

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Rubi [A]  time = 0.10857, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^5 c^2 \sqrt{c x^2} \log (a+b x)}{b^6 x}+\frac{a^4 c^2 \sqrt{c x^2}}{b^5}-\frac{a^3 c^2 x \sqrt{c x^2}}{2 b^4}+\frac{a^2 c^2 x^2 \sqrt{c x^2}}{3 b^3}-\frac{a c^2 x^3 \sqrt{c x^2}}{4 b^2}+\frac{c^2 x^4 \sqrt{c x^2}}{5 b} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^(5/2)/(a + b*x),x]

[Out]

(a^4*c^2*Sqrt[c*x^2])/b^5 - (a^3*c^2*x*Sqrt[c*x^2])/(2*b^4) + (a^2*c^2*x^2*Sqrt[
c*x^2])/(3*b^3) - (a*c^2*x^3*Sqrt[c*x^2])/(4*b^2) + (c^2*x^4*Sqrt[c*x^2])/(5*b)
- (a^5*c^2*Sqrt[c*x^2]*Log[a + b*x])/(b^6*x)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5} c^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{6} x} - \frac{a^{3} c^{2} \sqrt{c x^{2}} \int x\, dx}{b^{4} x} + \frac{a^{2} c^{2} x^{2} \sqrt{c x^{2}}}{3 b^{3}} - \frac{a c^{2} x^{3} \sqrt{c x^{2}}}{4 b^{2}} + \frac{c^{2} x^{4} \sqrt{c x^{2}}}{5 b} + \frac{c^{2} \sqrt{c x^{2}} \int a^{4}\, dx}{b^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**5*c**2*sqrt(c*x**2)*log(a + b*x)/(b**6*x) - a**3*c**2*sqrt(c*x**2)*Integral(
x, x)/(b**4*x) + a**2*c**2*x**2*sqrt(c*x**2)/(3*b**3) - a*c**2*x**3*sqrt(c*x**2)
/(4*b**2) + c**2*x**4*sqrt(c*x**2)/(5*b) + c**2*sqrt(c*x**2)*Integral(a**4, x)/(
b**5*x)

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Mathematica [A]  time = 0.0361824, size = 76, normalized size = 0.54 \[ \frac{c^3 x \left (b x \left (60 a^4-30 a^3 b x+20 a^2 b^2 x^2-15 a b^3 x^3+12 b^4 x^4\right )-60 a^5 \log (a+b x)\right )}{60 b^6 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c^3*x*(b*x*(60*a^4 - 30*a^3*b*x + 20*a^2*b^2*x^2 - 15*a*b^3*x^3 + 12*b^4*x^4) -
60*a^5*Log[a + b*x]))/(60*b^6*Sqrt[c*x^2])

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Maple [A]  time = 0.008, size = 74, normalized size = 0.5 \[ -{\frac{-12\,{b}^{5}{x}^{5}+15\,a{b}^{4}{x}^{4}-20\,{a}^{2}{b}^{3}{x}^{3}+30\,{a}^{3}{b}^{2}{x}^{2}+60\,{a}^{5}\ln \left ( bx+a \right ) -60\,{a}^{4}bx}{60\,{x}^{5}{b}^{6}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)/(b*x+a),x)

[Out]

-1/60*(c*x^2)^(5/2)*(-12*b^5*x^5+15*a*b^4*x^4-20*a^2*b^3*x^3+30*a^3*b^2*x^2+60*a
^5*ln(b*x+a)-60*a^4*b*x)/x^5/b^6

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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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213358, size = 123, normalized size = 0.87 \[ \frac{{\left (12 \, b^{5} c^{2} x^{5} - 15 \, a b^{4} c^{2} x^{4} + 20 \, a^{2} b^{3} c^{2} x^{3} - 30 \, a^{3} b^{2} c^{2} x^{2} + 60 \, a^{4} b c^{2} x - 60 \, a^{5} c^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{60 \, b^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(12*b^5*c^2*x^5 - 15*a*b^4*c^2*x^4 + 20*a^2*b^3*c^2*x^3 - 30*a^3*b^2*c^2*x^
2 + 60*a^4*b*c^2*x - 60*a^5*c^2*log(b*x + a))*sqrt(c*x^2)/(b^6*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.205415, size = 157, normalized size = 1.11 \[ -\frac{1}{60} \,{\left (\frac{60 \, a^{5} c^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{6}} - \frac{60 \, a^{5} c^{2}{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{6}} - \frac{12 \, b^{4} c^{2} x^{5}{\rm sign}\left (x\right ) - 15 \, a b^{3} c^{2} x^{4}{\rm sign}\left (x\right ) + 20 \, a^{2} b^{2} c^{2} x^{3}{\rm sign}\left (x\right ) - 30 \, a^{3} b c^{2} x^{2}{\rm sign}\left (x\right ) + 60 \, a^{4} c^{2} x{\rm sign}\left (x\right )}{b^{5}}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(60*a^5*c^2*ln(abs(b*x + a))*sign(x)/b^6 - 60*a^5*c^2*ln(abs(a))*sign(x)/b
^6 - (12*b^4*c^2*x^5*sign(x) - 15*a*b^3*c^2*x^4*sign(x) + 20*a^2*b^2*c^2*x^3*sig
n(x) - 30*a^3*b*c^2*x^2*sign(x) + 60*a^4*c^2*x*sign(x))/b^5)*sqrt(c)