3.1379 \(\int (a+b x) \sqrt{c+d x} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 b (c+d x)^{5/2}}{5 d^2}-\frac{2 (c+d x)^{3/2} (b c-a d)}{3 d^2} \]

[Out]

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

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Rubi [A]  time = 0.0439826, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 b (c+d x)^{5/2}}{5 d^2}-\frac{2 (c+d x)^{3/2} (b c-a d)}{3 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*Sqrt[c + d*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.76607, size = 37, normalized size = 0.88 \[ \frac{2 b \left (c + d x\right )^{\frac{5}{2}}}{5 d^{2}} + \frac{2 \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0307513, size = 30, normalized size = 0.71 \[ \frac{2 (c+d x)^{3/2} (5 a d-2 b c+3 b d x)}{15 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2)*(-2*b*c + 5*a*d + 3*b*d*x))/(15*d^2)

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Maple [A]  time = 0.004, size = 27, normalized size = 0.6 \[{\frac{6\,bdx+10\,ad-4\,bc}{15\,{d}^{2}} \left ( dx+c \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(d*x+c)^(1/2),x)

[Out]

2/15*(d*x+c)^(3/2)*(3*b*d*x+5*a*d-2*b*c)/d^2

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Maxima [A]  time = 1.3487, size = 45, normalized size = 1.07 \[ \frac{2 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} b - 5 \,{\left (b c - a d\right )}{\left (d x + c\right )}^{\frac{3}{2}}\right )}}{15 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*(d*x + c)^(5/2)*b - 5*(b*c - a*d)*(d*x + c)^(3/2))/d^2

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Fricas [A]  time = 0.208575, size = 62, normalized size = 1.48 \[ \frac{2 \,{\left (3 \, b d^{2} x^{2} - 2 \, b c^{2} + 5 \, a c d +{\left (b c d + 5 \, a d^{2}\right )} x\right )} \sqrt{d x + c}}{15 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*b*d^2*x^2 - 2*b*c^2 + 5*a*c*d + (b*c*d + 5*a*d^2)*x)*sqrt(d*x + c)/d^2

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Sympy [A]  time = 1.0942, size = 36, normalized size = 0.86 \[ \frac{2 \left (\frac{b \left (c + d x\right )^{\frac{5}{2}}}{5 d} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214511, size = 55, normalized size = 1.31 \[ \frac{2 \,{\left (5 \,{\left (d x + c\right )}^{\frac{3}{2}} a + \frac{{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} b}{d}\right )}}{15 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(5*(d*x + c)^(3/2)*a + (3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*b/d)/d