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

Optimal. Leaf size=41 \[ \frac{1}{6} a c^2 x^5 \sqrt{c x^2}+\frac{1}{7} b c^2 x^6 \sqrt{c x^2} \]

[Out]

(a*c^2*x^5*Sqrt[c*x^2])/6 + (b*c^2*x^6*Sqrt[c*x^2])/7

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Rubi [A]  time = 0.030896, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{6} a c^2 x^5 \sqrt{c x^2}+\frac{1}{7} b c^2 x^6 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^(5/2)*(a + b*x),x]

[Out]

(a*c^2*x^5*Sqrt[c*x^2])/6 + (b*c^2*x^6*Sqrt[c*x^2])/7

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Rubi in Sympy [A]  time = 6.0893, size = 36, normalized size = 0.88 \[ \frac{a c^{2} x^{5} \sqrt{c x^{2}}}{6} + \frac{b c^{2} x^{6} \sqrt{c x^{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**2*x**5*sqrt(c*x**2)/6 + b*c**2*x**6*sqrt(c*x**2)/7

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Mathematica [A]  time = 0.00902, size = 22, normalized size = 0.54 \[ \frac{1}{42} x \left (c x^2\right )^{5/2} (7 a+6 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*x^2)^(5/2)*(7*a + 6*b*x))/42

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Maple [A]  time = 0.005, size = 19, normalized size = 0.5 \[{\frac{x \left ( 6\,bx+7\,a \right ) }{42} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)*(b*x+a),x)

[Out]

1/42*x*(6*b*x+7*a)*(c*x^2)^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.205163, size = 38, normalized size = 0.93 \[ \frac{1}{42} \,{\left (6 \, b c^{2} x^{6} + 7 \, a c^{2} x^{5}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*(6*b*c^2*x^6 + 7*a*c^2*x^5)*sqrt(c*x^2)

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Sympy [A]  time = 5.143, size = 34, normalized size = 0.83 \[ \frac{a c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}}{6} + \frac{b c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**(5/2)*x*(x**2)**(5/2)/6 + b*c**(5/2)*x**2*(x**2)**(5/2)/7

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GIAC/XCAS [A]  time = 0.20549, size = 38, normalized size = 0.93 \[ \frac{1}{42} \,{\left (6 \, b c^{2} x^{7}{\rm sign}\left (x\right ) + 7 \, a c^{2} x^{6}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*(6*b*c^2*x^7*sign(x) + 7*a*c^2*x^6*sign(x))*sqrt(c)