3.340 \(\int \sqrt{d+e x} \left (b x+c x^2\right ) \, dx\)

Optimal. Leaf size=68 \[ -\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 d (d+e x)^{3/2} (c d-b e)}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3} \]

[Out]

(2*d*(c*d - b*e)*(d + e*x)^(3/2))/(3*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5
*e^3) + (2*c*(d + e*x)^(7/2))/(7*e^3)

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Rubi [A]  time = 0.0919436, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 (d+e x)^{5/2} (2 c d-b e)}{5 e^3}+\frac{2 d (d+e x)^{3/2} (c d-b e)}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]*(b*x + c*x^2),x]

[Out]

(2*d*(c*d - b*e)*(d + e*x)^(3/2))/(3*e^3) - (2*(2*c*d - b*e)*(d + e*x)^(5/2))/(5
*e^3) + (2*c*(d + e*x)^(7/2))/(7*e^3)

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Rubi in Sympy [A]  time = 13.3782, size = 63, normalized size = 0.93 \[ \frac{2 c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{3}} - \frac{2 d \left (d + e x\right )^{\frac{3}{2}} \left (b e - c d\right )}{3 e^{3}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)*(e*x+d)**(1/2),x)

[Out]

2*c*(d + e*x)**(7/2)/(7*e**3) - 2*d*(d + e*x)**(3/2)*(b*e - c*d)/(3*e**3) + 2*(d
+ e*x)**(5/2)*(b*e - 2*c*d)/(5*e**3)

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Mathematica [A]  time = 0.0453768, size = 50, normalized size = 0.74 \[ \frac{2 (d+e x)^{3/2} \left (7 b e (3 e x-2 d)+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]*(b*x + c*x^2),x]

[Out]

(2*(d + e*x)^(3/2)*(7*b*e*(-2*d + 3*e*x) + c*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(
105*e^3)

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Maple [A]  time = 0.006, size = 47, normalized size = 0.7 \[ -{\frac{-30\,c{e}^{2}{x}^{2}-42\,b{e}^{2}x+24\,cdex+28\,bde-16\,c{d}^{2}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)*(e*x+d)^(1/2),x)

[Out]

-2/105*(e*x+d)^(3/2)*(-15*c*e^2*x^2-21*b*e^2*x+12*c*d*e*x+14*b*d*e-8*c*d^2)/e^3

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Maxima [A]  time = 0.687198, size = 73, normalized size = 1.07 \[ \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} c - 21 \,{\left (2 \, c d - b e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (c d^{2} - b d e\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/105*(15*(e*x + d)^(7/2)*c - 21*(2*c*d - b*e)*(e*x + d)^(5/2) + 35*(c*d^2 - b*d
*e)*(e*x + d)^(3/2))/e^3

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Fricas [A]  time = 0.21725, size = 96, normalized size = 1.41 \[ \frac{2 \,{\left (15 \, c e^{3} x^{3} + 8 \, c d^{3} - 14 \, b d^{2} e + 3 \,{\left (c d e^{2} + 7 \, b e^{3}\right )} x^{2} -{\left (4 \, c d^{2} e - 7 \, b d e^{2}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/105*(15*c*e^3*x^3 + 8*c*d^3 - 14*b*d^2*e + 3*(c*d*e^2 + 7*b*e^3)*x^2 - (4*c*d^
2*e - 7*b*d*e^2)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 3.16186, size = 66, normalized size = 0.97 \[ \frac{2 \left (\frac{c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (b e - 2 c d\right )}{5 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (- b d e + c d^{2}\right )}{3 e^{2}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)*(e*x+d)**(1/2),x)

[Out]

2*(c*(d + e*x)**(7/2)/(7*e**2) + (d + e*x)**(5/2)*(b*e - 2*c*d)/(5*e**2) + (d +
e*x)**(3/2)*(-b*d*e + c*d**2)/(3*e**2))/e

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GIAC/XCAS [A]  time = 0.204982, size = 104, normalized size = 1.53 \[ \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} b e^{\left (-1\right )} +{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} c e^{\left (-14\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*sqrt(e*x + d),x, algorithm="giac")

[Out]

2/105*(7*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*b*e^(-1) + (15*(x*e + d)^(7/2
)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*c*e^(-14))*e^(
-1)