3.1391 \(\int (a+b x) (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 b (c+d x)^{7/2}}{7 d^2}-\frac{2 (c+d x)^{5/2} (b c-a d)}{5 d^2} \]

[Out]

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

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Rubi [A]  time = 0.042935, 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 \[ \frac{2 b (c+d x)^{7/2}}{7 d^2}-\frac{2 (c+d x)^{5/2} (b c-a d)}{5 d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(c + d*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.49445, size = 37, normalized size = 0.88 \[ \frac{2 b \left (c + d x\right )^{\frac{7}{2}}}{7 d^{2}} + \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (a d - b c\right )}{5 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(d*x+c)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0373612, size = 30, normalized size = 0.71 \[ \frac{2 (c+d x)^{5/2} (7 a d-2 b c+5 b d x)}{35 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2)*(-2*b*c + 7*a*d + 5*b*d*x))/(35*d^2)

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Maple [A]  time = 0.004, size = 27, normalized size = 0.6 \[{\frac{10\,bdx+14\,ad-4\,bc}{35\,{d}^{2}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(d*x+c)^(3/2),x)

[Out]

2/35*(d*x+c)^(5/2)*(5*b*d*x+7*a*d-2*b*c)/d^2

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Maxima [A]  time = 1.39211, size = 45, normalized size = 1.07 \[ \frac{2 \,{\left (5 \,{\left (d x + c\right )}^{\frac{7}{2}} b - 7 \,{\left (b c - a d\right )}{\left (d x + c\right )}^{\frac{5}{2}}\right )}}{35 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(3/2),x, algorithm="maxima")

[Out]

2/35*(5*(d*x + c)^(7/2)*b - 7*(b*c - a*d)*(d*x + c)^(5/2))/d^2

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Fricas [A]  time = 0.203142, size = 93, normalized size = 2.21 \[ \frac{2 \,{\left (5 \, b d^{3} x^{3} - 2 \, b c^{3} + 7 \, a c^{2} d +{\left (8 \, b c d^{2} + 7 \, a d^{3}\right )} x^{2} +{\left (b c^{2} d + 14 \, a c d^{2}\right )} x\right )} \sqrt{d x + c}}{35 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(3/2),x, algorithm="fricas")

[Out]

2/35*(5*b*d^3*x^3 - 2*b*c^3 + 7*a*c^2*d + (8*b*c*d^2 + 7*a*d^3)*x^2 + (b*c^2*d +
14*a*c*d^2)*x)*sqrt(d*x + c)/d^2

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Sympy [A]  time = 0.867061, size = 146, normalized size = 3.48 \[ \begin{cases} \frac{2 a c^{2} \sqrt{c + d x}}{5 d} + \frac{4 a c x \sqrt{c + d x}}{5} + \frac{2 a d x^{2} \sqrt{c + d x}}{5} - \frac{4 b c^{3} \sqrt{c + d x}}{35 d^{2}} + \frac{2 b c^{2} x \sqrt{c + d x}}{35 d} + \frac{16 b c x^{2} \sqrt{c + d x}}{35} + \frac{2 b d x^{3} \sqrt{c + d x}}{7} & \text{for}\: d \neq 0 \\c^{\frac{3}{2}} \left (a x + \frac{b x^{2}}{2}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(d*x+c)**(3/2),x)

[Out]

Piecewise((2*a*c**2*sqrt(c + d*x)/(5*d) + 4*a*c*x*sqrt(c + d*x)/5 + 2*a*d*x**2*s
qrt(c + d*x)/5 - 4*b*c**3*sqrt(c + d*x)/(35*d**2) + 2*b*c**2*x*sqrt(c + d*x)/(35
*d) + 16*b*c*x**2*sqrt(c + d*x)/35 + 2*b*d*x**3*sqrt(c + d*x)/7, Ne(d, 0)), (c**
(3/2)*(a*x + b*x**2/2), True))

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GIAC/XCAS [A]  time = 0.221841, size = 153, normalized size = 3.64 \[ \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{3}{2}} a c + 7 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a + \frac{7 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} b c}{d} + \frac{{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{12} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{12} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{12}\right )} b}{d^{13}}\right )}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(d*x + c)^(3/2),x, algorithm="giac")

[Out]

2/105*(35*(d*x + c)^(3/2)*a*c + 7*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a +
7*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*b*c/d + (15*(d*x + c)^(7/2)*d^12 - 4
2*(d*x + c)^(5/2)*c*d^12 + 35*(d*x + c)^(3/2)*c^2*d^12)*b/d^13)/d