3.1377 \(\int (a+b x)^3 \sqrt{c+d x} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0355149, 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} \[ -\frac{6 b^2 (c+d x)^{7/2} (b c-a d)}{7 d^4}+\frac{6 b (c+d x)^{5/2} (b c-a d)^2}{5 d^4}-\frac{2 (c+d x)^{3/2} (b c-a d)^3}{3 d^4}+\frac{2 b^3 (c+d x)^{9/2}}{9 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*Sqrt[c + d*x],x]

[Out]

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

Mathematica [A]  time = 0.0604461, size = 79, normalized size = 0.79 \[ \frac{2 (c+d x)^{3/2} \left (-135 b^2 (c+d x)^2 (b c-a d)+189 b (c+d x) (b c-a d)^2-105 (b c-a d)^3+35 b^3 (c+d x)^3\right )}{315 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(3/2)*(-105*(b*c - a*d)^3 + 189*b*(b*c - a*d)^2*(c + d*x) - 135*b^2*(b*c - a*d)*(c + d*x)^2 + 35*
b^3*(c + d*x)^3))/(315*d^4)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 116, normalized size = 1.2 \begin{align*}{\frac{70\,{b}^{3}{x}^{3}{d}^{3}+270\,a{b}^{2}{d}^{3}{x}^{2}-60\,{b}^{3}c{d}^{2}{x}^{2}+378\,{a}^{2}b{d}^{3}x-216\,a{b}^{2}c{d}^{2}x+48\,{b}^{3}{c}^{2}dx+210\,{a}^{3}{d}^{3}-252\,{a}^{2}bc{d}^{2}+144\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{315\,{d}^{4}} \left ( dx+c \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(d*x+c)^(1/2),x)

[Out]

2/315*(d*x+c)^(3/2)*(35*b^3*d^3*x^3+135*a*b^2*d^3*x^2-30*b^3*c*d^2*x^2+189*a^2*b*d^3*x-108*a*b^2*c*d^2*x+24*b^
3*c^2*d*x+105*a^3*d^3-126*a^2*b*c*d^2+72*a*b^2*c^2*d-16*b^3*c^3)/d^4

________________________________________________________________________________________

Maxima [A]  time = 0.967357, size = 159, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{3} - 135 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 189 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left (d x + c\right )}^{\frac{5}{2}} - 105 \,{\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{3}{2}}\right )}}{315 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*(d*x + c)^(9/2)*b^3 - 135*(b^3*c - a*b^2*d)*(d*x + c)^(7/2) + 189*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2
)*(d*x + c)^(5/2) - 105*(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)^(3/2))/d^4

________________________________________________________________________________________

Fricas [A]  time = 2.05845, size = 359, normalized size = 3.59 \begin{align*} \frac{2 \,{\left (35 \, b^{3} d^{4} x^{4} - 16 \, b^{3} c^{4} + 72 \, a b^{2} c^{3} d - 126 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3} + 5 \,{\left (b^{3} c d^{3} + 27 \, a b^{2} d^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} c^{2} d^{2} - 9 \, a b^{2} c d^{3} - 63 \, a^{2} b d^{4}\right )} x^{2} +{\left (8 \, b^{3} c^{3} d - 36 \, a b^{2} c^{2} d^{2} + 63 \, a^{2} b c d^{3} + 105 \, a^{3} d^{4}\right )} x\right )} \sqrt{d x + c}}{315 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^3*d^4*x^4 - 16*b^3*c^4 + 72*a*b^2*c^3*d - 126*a^2*b*c^2*d^2 + 105*a^3*c*d^3 + 5*(b^3*c*d^3 + 27*a*
b^2*d^4)*x^3 - 3*(2*b^3*c^2*d^2 - 9*a*b^2*c*d^3 - 63*a^2*b*d^4)*x^2 + (8*b^3*c^3*d - 36*a*b^2*c^2*d^2 + 63*a^2
*b*c*d^3 + 105*a^3*d^4)*x)*sqrt(d*x + c)/d^4

________________________________________________________________________________________

Sympy [A]  time = 2.36295, size = 146, normalized size = 1.46 \begin{align*} \frac{2 \left (\frac{b^{3} \left (c + d x\right )^{\frac{9}{2}}}{9 d^{3}} + \frac{\left (c + d x\right )^{\frac{7}{2}} \left (3 a b^{2} d - 3 b^{3} c\right )}{7 d^{3}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (3 a^{2} b d^{2} - 6 a b^{2} c d + 3 b^{3} c^{2}\right )}{5 d^{3}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{3 d^{3}}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(d*x+c)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.06367, size = 188, normalized size = 1.88 \begin{align*} \frac{2 \,{\left (105 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{3} + \frac{63 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a^{2} b}{d} + \frac{9 \,{\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}}{d^{2}} + \frac{{\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}}{d^{3}}\right )}}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/315*(105*(d*x + c)^(3/2)*a^3 + 63*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^2*b/d + 9*(15*(d*x + c)^(7/2)
- 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a*b^2/d^2 + (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/d^3)/d