3.1568 \(\int (b+2 c x) \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=18 \[ \frac{2}{7} \left (a+b x+c x^2\right )^{7/2} \]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/7

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Rubi [A]  time = 0.0103703, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{2}{7} \left (a+b x+c x^2\right )^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x + c*x^2)^(7/2))/7

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Rubi in Sympy [A]  time = 3.79022, size = 15, normalized size = 0.83 \[ \frac{2 \left (a + b x + c x^{2}\right )^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a + b*x + c*x**2)**(7/2)/7

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Mathematica [A]  time = 0.0339204, size = 17, normalized size = 0.94 \[ \frac{2}{7} (a+x (b+c x))^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + x*(b + c*x))^(7/2))/7

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Maple [A]  time = 0.004, size = 15, normalized size = 0.8 \[{\frac{2}{7} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(c*x^2+b*x+a)^(7/2)

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Maxima [A]  time = 0.700487, size = 19, normalized size = 1.06 \[ \frac{2}{7} \,{\left (c x^{2} + b x + a\right )}^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(c*x^2 + b*x + a)^(7/2)

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Fricas [A]  time = 0.325072, size = 116, normalized size = 6.44 \[ \frac{2}{7} \,{\left (c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{2}\right )} \sqrt{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)
*x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^2)*sqrt(c*x^2 + b*x + a)

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Sympy [A]  time = 8.40931, size = 243, normalized size = 13.5 \[ \frac{2 a^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a^{2} b x \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a^{2} c x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a b^{2} x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{12 a b c x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 a c^{2} x^{4} \sqrt{a + b x + c x^{2}}}{7} + \frac{2 b^{3} x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 b^{2} c x^{4} \sqrt{a + b x + c x^{2}}}{7} + \frac{6 b c^{2} x^{5} \sqrt{a + b x + c x^{2}}}{7} + \frac{2 c^{3} x^{6} \sqrt{a + b x + c x^{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3*sqrt(a + b*x + c*x**2)/7 + 6*a**2*b*x*sqrt(a + b*x + c*x**2)/7 + 6*a**2*c
*x**2*sqrt(a + b*x + c*x**2)/7 + 6*a*b**2*x**2*sqrt(a + b*x + c*x**2)/7 + 12*a*b
*c*x**3*sqrt(a + b*x + c*x**2)/7 + 6*a*c**2*x**4*sqrt(a + b*x + c*x**2)/7 + 2*b*
*3*x**3*sqrt(a + b*x + c*x**2)/7 + 6*b**2*c*x**4*sqrt(a + b*x + c*x**2)/7 + 6*b*
c**2*x**5*sqrt(a + b*x + c*x**2)/7 + 2*c**3*x**6*sqrt(a + b*x + c*x**2)/7

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GIAC/XCAS [A]  time = 0.275675, size = 146, normalized size = 8.11 \[ \frac{2}{7} \,{\left (a^{3} +{\left (3 \, a^{2} b +{\left ({\left ({\left ({\left (c^{3} x + 3 \, b c^{2}\right )} x + \frac{3 \,{\left (b^{2} c^{7} + a c^{8}\right )}}{c^{6}}\right )} x + \frac{b^{3} c^{6} + 6 \, a b c^{7}}{c^{6}}\right )} x + \frac{3 \,{\left (a b^{2} c^{6} + a^{2} c^{7}\right )}}{c^{6}}\right )} x\right )} x\right )} \sqrt{c x^{2} + b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(a^3 + (3*a^2*b + ((((c^3*x + 3*b*c^2)*x + 3*(b^2*c^7 + a*c^8)/c^6)*x + (b^3
*c^6 + 6*a*b*c^7)/c^6)*x + 3*(a*b^2*c^6 + a^2*c^7)/c^6)*x)*x)*sqrt(c*x^2 + b*x +
a)