3.1389 \(\int (a+b x)^3 (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 b^2 (c+d x)^{9/2} (b c-a d)}{3 d^4}+\frac{6 b (c+d x)^{7/2} (b c-a d)^2}{7 d^4}-\frac{2 (c+d x)^{5/2} (b c-a d)^3}{5 d^4}+\frac{2 b^3 (c+d x)^{11/2}}{11 d^4} \]

[Out]

(-2*(b*c - a*d)^3*(c + d*x)^(5/2))/(5*d^4) + (6*b*(b*c - a*d)^2*(c + d*x)^(7/2))/(7*d^4) - (2*b^2*(b*c - a*d)*
(c + d*x)^(9/2))/(3*d^4) + (2*b^3*(c + d*x)^(11/2))/(11*d^4)

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Rubi [A]  time = 0.0338238, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {43} \[ -\frac{2 b^2 (c+d x)^{9/2} (b c-a d)}{3 d^4}+\frac{6 b (c+d x)^{7/2} (b c-a d)^2}{7 d^4}-\frac{2 (c+d x)^{5/2} (b c-a d)^3}{5 d^4}+\frac{2 b^3 (c+d x)^{11/2}}{11 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(c + d*x)^(3/2),x]

[Out]

(-2*(b*c - a*d)^3*(c + d*x)^(5/2))/(5*d^4) + (6*b*(b*c - a*d)^2*(c + d*x)^(7/2))/(7*d^4) - (2*b^2*(b*c - a*d)*
(c + d*x)^(9/2))/(3*d^4) + (2*b^3*(c + d*x)^(11/2))/(11*d^4)

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)^3 (c+d x)^{3/2} \, dx &=\int \left (\frac{(-b c+a d)^3 (c+d x)^{3/2}}{d^3}+\frac{3 b (b c-a d)^2 (c+d x)^{5/2}}{d^3}-\frac{3 b^2 (b c-a d) (c+d x)^{7/2}}{d^3}+\frac{b^3 (c+d x)^{9/2}}{d^3}\right ) \, dx\\ &=-\frac{2 (b c-a d)^3 (c+d x)^{5/2}}{5 d^4}+\frac{6 b (b c-a d)^2 (c+d x)^{7/2}}{7 d^4}-\frac{2 b^2 (b c-a d) (c+d x)^{9/2}}{3 d^4}+\frac{2 b^3 (c+d x)^{11/2}}{11 d^4}\\ \end{align*}

Mathematica [A]  time = 0.0661977, size = 79, normalized size = 0.79 \[ \frac{2 (c+d x)^{5/2} \left (-385 b^2 (c+d x)^2 (b c-a d)+495 b (c+d x) (b c-a d)^2-231 (b c-a d)^3+105 b^3 (c+d x)^3\right )}{1155 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(5/2)*(-231*(b*c - a*d)^3 + 495*b*(b*c - a*d)^2*(c + d*x) - 385*b^2*(b*c - a*d)*(c + d*x)^2 + 105
*b^3*(c + d*x)^3))/(1155*d^4)

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Maple [A]  time = 0.006, size = 116, normalized size = 1.2 \begin{align*}{\frac{210\,{b}^{3}{x}^{3}{d}^{3}+770\,a{b}^{2}{d}^{3}{x}^{2}-140\,{b}^{3}c{d}^{2}{x}^{2}+990\,{a}^{2}b{d}^{3}x-440\,a{b}^{2}c{d}^{2}x+80\,{b}^{3}{c}^{2}dx+462\,{a}^{3}{d}^{3}-396\,{a}^{2}bc{d}^{2}+176\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{1155\,{d}^{4}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(d*x+c)^(3/2),x)

[Out]

2/1155*(d*x+c)^(5/2)*(105*b^3*d^3*x^3+385*a*b^2*d^3*x^2-70*b^3*c*d^2*x^2+495*a^2*b*d^3*x-220*a*b^2*c*d^2*x+40*
b^3*c^2*d*x+231*a^3*d^3-198*a^2*b*c*d^2+88*a*b^2*c^2*d-16*b^3*c^3)/d^4

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Maxima [A]  time = 0.948524, size = 159, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (105 \,{\left (d x + c\right )}^{\frac{11}{2}} b^{3} - 385 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}^{\frac{9}{2}} + 495 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left (d x + c\right )}^{\frac{7}{2}} - 231 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x + c\right )}^{\frac{5}{2}}\right )}}{1155 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*(d*x + c)^(11/2)*b^3 - 385*(b^3*c - a*b^2*d)*(d*x + c)^(9/2) + 495*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*
d^2)*(d*x + c)^(7/2) - 231*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*(d*x + c)^(5/2))/d^4

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Fricas [B]  time = 1.98509, size = 474, normalized size = 4.74 \begin{align*} \frac{2 \,{\left (105 \, b^{3} d^{5} x^{5} - 16 \, b^{3} c^{5} + 88 \, a b^{2} c^{4} d - 198 \, a^{2} b c^{3} d^{2} + 231 \, a^{3} c^{2} d^{3} + 35 \,{\left (4 \, b^{3} c d^{4} + 11 \, a b^{2} d^{5}\right )} x^{4} + 5 \,{\left (b^{3} c^{2} d^{3} + 110 \, a b^{2} c d^{4} + 99 \, a^{2} b d^{5}\right )} x^{3} - 3 \,{\left (2 \, b^{3} c^{3} d^{2} - 11 \, a b^{2} c^{2} d^{3} - 264 \, a^{2} b c d^{4} - 77 \, a^{3} d^{5}\right )} x^{2} +{\left (8 \, b^{3} c^{4} d - 44 \, a b^{2} c^{3} d^{2} + 99 \, a^{2} b c^{2} d^{3} + 462 \, a^{3} c d^{4}\right )} x\right )} \sqrt{d x + c}}{1155 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1155*(105*b^3*d^5*x^5 - 16*b^3*c^5 + 88*a*b^2*c^4*d - 198*a^2*b*c^3*d^2 + 231*a^3*c^2*d^3 + 35*(4*b^3*c*d^4
+ 11*a*b^2*d^5)*x^4 + 5*(b^3*c^2*d^3 + 110*a*b^2*c*d^4 + 99*a^2*b*d^5)*x^3 - 3*(2*b^3*c^3*d^2 - 11*a*b^2*c^2*d
^3 - 264*a^2*b*c*d^4 - 77*a^3*d^5)*x^2 + (8*b^3*c^4*d - 44*a*b^2*c^3*d^2 + 99*a^2*b*c^2*d^3 + 462*a^3*c*d^4)*x
)*sqrt(d*x + c)/d^4

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Sympy [A]  time = 10.9798, size = 386, normalized size = 3.86 \begin{align*} a^{3} c \left (\begin{cases} \sqrt{c} x & \text{for}\: d = 0 \\\frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{3} \left (- \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d} + \frac{6 a^{2} b c \left (- \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} + \frac{6 a^{2} b \left (\frac{c^{2} \left (c + d x\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + d x\right )^{\frac{5}{2}}}{5} + \frac{\left (c + d x\right )^{\frac{7}{2}}}{7}\right )}{d^{2}} + \frac{6 a b^{2} c \left (\frac{c^{2} \left (c + d x\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + d x\right )^{\frac{5}{2}}}{5} + \frac{\left (c + d x\right )^{\frac{7}{2}}}{7}\right )}{d^{3}} + \frac{6 a b^{2} \left (- \frac{c^{3} \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + d x\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + d x\right )^{\frac{7}{2}}}{7} + \frac{\left (c + d x\right )^{\frac{9}{2}}}{9}\right )}{d^{3}} + \frac{2 b^{3} c \left (- \frac{c^{3} \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + d x\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + d x\right )^{\frac{7}{2}}}{7} + \frac{\left (c + d x\right )^{\frac{9}{2}}}{9}\right )}{d^{4}} + \frac{2 b^{3} \left (\frac{c^{4} \left (c + d x\right )^{\frac{3}{2}}}{3} - \frac{4 c^{3} \left (c + d x\right )^{\frac{5}{2}}}{5} + \frac{6 c^{2} \left (c + d x\right )^{\frac{7}{2}}}{7} - \frac{4 c \left (c + d x\right )^{\frac{9}{2}}}{9} + \frac{\left (c + d x\right )^{\frac{11}{2}}}{11}\right )}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*c*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 2*a**3*(-c*(c + d*x)**(3/2)/3 + (c
+ d*x)**(5/2)/5)/d + 6*a**2*b*c*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 6*a**2*b*(c**2*(c + d*x)*
*(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**2 + 6*a*b**2*c*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c +
d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**3 + 6*a*b**2*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5 - 3*
c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**3 + 2*b**3*c*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)
/5 - 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**4 + 2*b**3*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)**(
5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2)/9 + (c + d*x)**(11/2)/11)/d**4

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Giac [B]  time = 1.06874, size = 440, normalized size = 4.4 \begin{align*} \frac{2 \,{\left (1155 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} c + 231 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a^{3} + \frac{693 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a^{2} b c}{d} + \frac{99 \,{\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 )} a b^{2} c}{d^{2}} + \frac{99 \,{\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 )} a^{2} b}{d} + \frac{11 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3}\right )} b^{3} c}{d^{3}} + \frac{33 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3}\right )} a b^{2}}{d^{2}} + \frac{{\left (315 \,{\left (d x + c\right )}^{\frac{11}{2}} - 1540 \,{\left (d x + c\right )}^{\frac{9}{2}} c + 2970 \,{\left (d x + c\right )}^{\frac{7}{2}} c^{2} - 2772 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{3} + 1155 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{4}\right )} b^{3}}{d^{3}}\right )}}{3465 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(1155*(d*x + c)^(3/2)*a^3*c + 231*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^3 + 693*(3*(d*x + c)^(5/2
) - 5*(d*x + c)^(3/2)*c)*a^2*b*c/d + 99*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a
*b^2*c/d^2 + 99*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a^2*b/d + 11*(35*(d*x + c
)^(9/2) - 135*(d*x + c)^(7/2)*c + 189*(d*x + c)^(5/2)*c^2 - 105*(d*x + c)^(3/2)*c^3)*b^3*c/d^3 + 33*(35*(d*x +
c)^(9/2) - 135*(d*x + c)^(7/2)*c + 189*(d*x + c)^(5/2)*c^2 - 105*(d*x + c)^(3/2)*c^3)*a*b^2/d^2 + (315*(d*x +
c)^(11/2) - 1540*(d*x + c)^(9/2)*c + 2970*(d*x + c)^(7/2)*c^2 - 2772*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/
2)*c^4)*b^3/d^3)/d