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 = 0.0139126, 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, Rules used = {43} \[ \frac{2 b (c+d x)^{7/2}}{7 d^2}-\frac{2 (c+d x)^{5/2} (b c-a d)}{5 d^2} \]

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)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (c+d x)^{3/2} \, dx &=\int \left (\frac{(-b c+a d) (c+d x)^{3/2}}{d}+\frac{b (c+d x)^{5/2}}{d}\right ) \, dx\\ &=-\frac{2 (b c-a d) (c+d x)^{5/2}}{5 d^2}+\frac{2 b (c+d x)^{7/2}}{7 d^2}\\ \end{align*}

Mathematica [A]  time = 0.0202293, 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 = 0.002, size = 27, normalized size = 0.6 \begin{align*}{\frac{10\,bdx+14\,ad-4\,bc}{35\,{d}^{2}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \end{align*}

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 = 0.948696, size = 45, normalized size = 1.07 \begin{align*} \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}} \end{align*}

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 [B]  time = 1.94714, size = 155, normalized size = 3.69 \begin{align*} \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}} \end{align*}

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 = 0.681699, size = 146, normalized size = 3.48 \begin{align*} \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} \end{align*}

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*sqrt(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 [B]  time = 1.04931, size = 140, normalized size = 3.33 \begin{align*} \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}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2}\right )} b}{d}\right )}}{105 \, d} \end{align*}

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) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*b/d)/d