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

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

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Rubi [A]  time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.79 \begin {gather*} \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} \end {gather*}

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)

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IntegrateAlgebraic [A]  time = 0.05, size = 132, normalized size = 1.32 \begin {gather*} \frac {2 (c+d x)^{3/2} \left (105 a^3 d^3+189 a^2 b d^2 (c+d x)-315 a^2 b c d^2+315 a b^2 c^2 d+135 a b^2 d (c+d x)^2-378 a b^2 c d (c+d x)-105 b^3 c^3+189 b^3 c^2 (c+d x)+35 b^3 (c+d x)^3-135 b^3 c (c+d x)^2\right )}{315 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.16, size = 164, normalized size = 1.64 \begin {gather*} \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 {gather*}

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

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giac [B]  time = 1.27, size = 322, normalized size = 3.22 \begin {gather*} \frac {2 \, {\left (315 \, \sqrt {d x + c} a^{3} c + 105 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} + \frac {315 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} b c}{d} + \frac {63 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b^{2} c}{d^{2}} + \frac {63 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a^{2} b}{d} + \frac {9 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{3} c}{d^{3}} + \frac {27 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a b^{2}}{d^{2}} + \frac {{\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b^{3}}{d^{3}}\right )}}{315 \, d} \end {gather*}

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*(315*sqrt(d*x + c)*a^3*c + 105*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^3 + 315*((d*x + c)^(3/2) - 3*sqrt
(d*x + c)*c)*a^2*b*c/d + 63*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a*b^2*c/d^2 + 63
*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^2*b/d + 9*(5*(d*x + c)^(7/2) - 21*(d*x +
c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*b^3*c/d^3 + 27*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(
5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a*b^2/d^2 + (35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*
c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*b^3/d^3)/d

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maple [A]  time = 0.00, size = 116, normalized size = 1.16 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (35 b^{3} x^{3} d^{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}\right )}{315 d^{4}} \end {gather*}

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

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maxima [A]  time = 1.37, size = 118, normalized size = 1.18 \begin {gather*} \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 {gather*}

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

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mupad [B]  time = 0.07, size = 87, normalized size = 0.87 \begin {gather*} \frac {2\,b^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{3/2}}{3\,d^4}+\frac {6\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{5/2}}{5\,d^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^3*(c + d*x)^(1/2),x)

[Out]

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

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sympy [A]  time = 3.34, size = 146, normalized size = 1.46 \begin {gather*} \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 {gather*}

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

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