3.1375 \(\int (a+b x)^5 \sqrt{c+d x} \, dx\)

Optimal. Leaf size=156 \[ -\frac{10 b^4 (c+d x)^{11/2} (b c-a d)}{11 d^6}+\frac{20 b^3 (c+d x)^{9/2} (b c-a d)^2}{9 d^6}-\frac{20 b^2 (c+d x)^{7/2} (b c-a d)^3}{7 d^6}+\frac{2 b (c+d x)^{5/2} (b c-a d)^4}{d^6}-\frac{2 (c+d x)^{3/2} (b c-a d)^5}{3 d^6}+\frac{2 b^5 (c+d x)^{13/2}}{13 d^6} \]

[Out]

(-2*(b*c - a*d)^5*(c + d*x)^(3/2))/(3*d^6) + (2*b*(b*c - a*d)^4*(c + d*x)^(5/2))/d^6 - (20*b^2*(b*c - a*d)^3*(
c + d*x)^(7/2))/(7*d^6) + (20*b^3*(b*c - a*d)^2*(c + d*x)^(9/2))/(9*d^6) - (10*b^4*(b*c - a*d)*(c + d*x)^(11/2
))/(11*d^6) + (2*b^5*(c + d*x)^(13/2))/(13*d^6)

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Rubi [A]  time = 0.0627983, antiderivative size = 156, 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} \[ -\frac{10 b^4 (c+d x)^{11/2} (b c-a d)}{11 d^6}+\frac{20 b^3 (c+d x)^{9/2} (b c-a d)^2}{9 d^6}-\frac{20 b^2 (c+d x)^{7/2} (b c-a d)^3}{7 d^6}+\frac{2 b (c+d x)^{5/2} (b c-a d)^4}{d^6}-\frac{2 (c+d x)^{3/2} (b c-a d)^5}{3 d^6}+\frac{2 b^5 (c+d x)^{13/2}}{13 d^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5*Sqrt[c + d*x],x]

[Out]

(-2*(b*c - a*d)^5*(c + d*x)^(3/2))/(3*d^6) + (2*b*(b*c - a*d)^4*(c + d*x)^(5/2))/d^6 - (20*b^2*(b*c - a*d)^3*(
c + d*x)^(7/2))/(7*d^6) + (20*b^3*(b*c - a*d)^2*(c + d*x)^(9/2))/(9*d^6) - (10*b^4*(b*c - a*d)*(c + d*x)^(11/2
))/(11*d^6) + (2*b^5*(c + d*x)^(13/2))/(13*d^6)

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)^5 \sqrt{c+d x} \, dx &=\int \left (\frac{(-b c+a d)^5 \sqrt{c+d x}}{d^5}+\frac{5 b (b c-a d)^4 (c+d x)^{3/2}}{d^5}-\frac{10 b^2 (b c-a d)^3 (c+d x)^{5/2}}{d^5}+\frac{10 b^3 (b c-a d)^2 (c+d x)^{7/2}}{d^5}-\frac{5 b^4 (b c-a d) (c+d x)^{9/2}}{d^5}+\frac{b^5 (c+d x)^{11/2}}{d^5}\right ) \, dx\\ &=-\frac{2 (b c-a d)^5 (c+d x)^{3/2}}{3 d^6}+\frac{2 b (b c-a d)^4 (c+d x)^{5/2}}{d^6}-\frac{20 b^2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^6}+\frac{20 b^3 (b c-a d)^2 (c+d x)^{9/2}}{9 d^6}-\frac{10 b^4 (b c-a d) (c+d x)^{11/2}}{11 d^6}+\frac{2 b^5 (c+d x)^{13/2}}{13 d^6}\\ \end{align*}

Mathematica [A]  time = 0.138097, size = 123, normalized size = 0.79 \[ \frac{2 (c+d x)^{3/2} \left (-12870 b^2 (c+d x)^2 (b c-a d)^3+10010 b^3 (c+d x)^3 (b c-a d)^2-4095 b^4 (c+d x)^4 (b c-a d)+9009 b (c+d x) (b c-a d)^4-3003 (b c-a d)^5+693 b^5 (c+d x)^5\right )}{9009 d^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5*Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2)*(-3003*(b*c - a*d)^5 + 9009*b*(b*c - a*d)^4*(c + d*x) - 12870*b^2*(b*c - a*d)^3*(c + d*x)^2
+ 10010*b^3*(b*c - a*d)^2*(c + d*x)^3 - 4095*b^4*(b*c - a*d)*(c + d*x)^4 + 693*b^5*(c + d*x)^5))/(9009*d^6)

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Maple [B]  time = 0.005, size = 273, normalized size = 1.8 \begin{align*}{\frac{1386\,{b}^{5}{x}^{5}{d}^{5}+8190\,a{b}^{4}{d}^{5}{x}^{4}-1260\,{b}^{5}c{d}^{4}{x}^{4}+20020\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}-7280\,a{b}^{4}c{d}^{4}{x}^{3}+1120\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}+25740\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}-17160\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}+6240\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}-960\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+18018\,{a}^{4}b{d}^{5}x-20592\,{a}^{3}{b}^{2}c{d}^{4}x+13728\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x-4992\,a{b}^{4}{c}^{3}{d}^{2}x+768\,{b}^{5}{c}^{4}dx+6006\,{a}^{5}{d}^{5}-12012\,{a}^{4}bc{d}^{4}+13728\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-9152\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+3328\,a{b}^{4}{c}^{4}d-512\,{b}^{5}{c}^{5}}{9009\,{d}^{6}} \left ( dx+c \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5*(d*x+c)^(1/2),x)

[Out]

2/9009*(d*x+c)^(3/2)*(693*b^5*d^5*x^5+4095*a*b^4*d^5*x^4-630*b^5*c*d^4*x^4+10010*a^2*b^3*d^5*x^3-3640*a*b^4*c*
d^4*x^3+560*b^5*c^2*d^3*x^3+12870*a^3*b^2*d^5*x^2-8580*a^2*b^3*c*d^4*x^2+3120*a*b^4*c^2*d^3*x^2-480*b^5*c^3*d^
2*x^2+9009*a^4*b*d^5*x-10296*a^3*b^2*c*d^4*x+6864*a^2*b^3*c^2*d^3*x-2496*a*b^4*c^3*d^2*x+384*b^5*c^4*d*x+3003*
a^5*d^5-6006*a^4*b*c*d^4+6864*a^3*b^2*c^2*d^3-4576*a^2*b^3*c^3*d^2+1664*a*b^4*c^4*d-256*b^5*c^5)/d^6

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Maxima [A]  time = 0.972099, size = 350, normalized size = 2.24 \begin{align*} \frac{2 \,{\left (693 \,{\left (d x + c\right )}^{\frac{13}{2}} b^{5} - 4095 \,{\left (b^{5} c - a b^{4} d\right )}{\left (d x + c\right )}^{\frac{11}{2}} + 10010 \,{\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )}{\left (d x + c\right )}^{\frac{9}{2}} - 12870 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 9009 \,{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}{\left (d x + c\right )}^{\frac{5}{2}} - 3003 \,{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )}{\left (d x + c\right )}^{\frac{3}{2}}\right )}}{9009 \, d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/9009*(693*(d*x + c)^(13/2)*b^5 - 4095*(b^5*c - a*b^4*d)*(d*x + c)^(11/2) + 10010*(b^5*c^2 - 2*a*b^4*c*d + a^
2*b^3*d^2)*(d*x + c)^(9/2) - 12870*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*(d*x + c)^(7/2) +
9009*(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*(d*x + c)^(5/2) - 3003*(b^5*
c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*(d*x + c)^(3/2))/d^6

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Fricas [B]  time = 2.10166, size = 761, normalized size = 4.88 \begin{align*} \frac{2 \,{\left (693 \, b^{5} d^{6} x^{6} - 256 \, b^{5} c^{6} + 1664 \, a b^{4} c^{5} d - 4576 \, a^{2} b^{3} c^{4} d^{2} + 6864 \, a^{3} b^{2} c^{3} d^{3} - 6006 \, a^{4} b c^{2} d^{4} + 3003 \, a^{5} c d^{5} + 63 \,{\left (b^{5} c d^{5} + 65 \, a b^{4} d^{6}\right )} x^{5} - 35 \,{\left (2 \, b^{5} c^{2} d^{4} - 13 \, a b^{4} c d^{5} - 286 \, a^{2} b^{3} d^{6}\right )} x^{4} + 10 \,{\left (8 \, b^{5} c^{3} d^{3} - 52 \, a b^{4} c^{2} d^{4} + 143 \, a^{2} b^{3} c d^{5} + 1287 \, a^{3} b^{2} d^{6}\right )} x^{3} - 3 \,{\left (32 \, b^{5} c^{4} d^{2} - 208 \, a b^{4} c^{3} d^{3} + 572 \, a^{2} b^{3} c^{2} d^{4} - 858 \, a^{3} b^{2} c d^{5} - 3003 \, a^{4} b d^{6}\right )} x^{2} +{\left (128 \, b^{5} c^{5} d - 832 \, a b^{4} c^{4} d^{2} + 2288 \, a^{2} b^{3} c^{3} d^{3} - 3432 \, a^{3} b^{2} c^{2} d^{4} + 3003 \, a^{4} b c d^{5} + 3003 \, a^{5} d^{6}\right )} x\right )} \sqrt{d x + c}}{9009 \, d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/9009*(693*b^5*d^6*x^6 - 256*b^5*c^6 + 1664*a*b^4*c^5*d - 4576*a^2*b^3*c^4*d^2 + 6864*a^3*b^2*c^3*d^3 - 6006*
a^4*b*c^2*d^4 + 3003*a^5*c*d^5 + 63*(b^5*c*d^5 + 65*a*b^4*d^6)*x^5 - 35*(2*b^5*c^2*d^4 - 13*a*b^4*c*d^5 - 286*
a^2*b^3*d^6)*x^4 + 10*(8*b^5*c^3*d^3 - 52*a*b^4*c^2*d^4 + 143*a^2*b^3*c*d^5 + 1287*a^3*b^2*d^6)*x^3 - 3*(32*b^
5*c^4*d^2 - 208*a*b^4*c^3*d^3 + 572*a^2*b^3*c^2*d^4 - 858*a^3*b^2*c*d^5 - 3003*a^4*b*d^6)*x^2 + (128*b^5*c^5*d
- 832*a*b^4*c^4*d^2 + 2288*a^2*b^3*c^3*d^3 - 3432*a^3*b^2*c^2*d^4 + 3003*a^4*b*c*d^5 + 3003*a^5*d^6)*x)*sqrt(
d*x + c)/d^6

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Sympy [B]  time = 3.75216, size = 314, normalized size = 2.01 \begin{align*} \frac{2 \left (\frac{b^{5} \left (c + d x\right )^{\frac{13}{2}}}{13 d^{5}} + \frac{\left (c + d x\right )^{\frac{11}{2}} \left (5 a b^{4} d - 5 b^{5} c\right )}{11 d^{5}} + \frac{\left (c + d x\right )^{\frac{9}{2}} \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{9 d^{5}} + \frac{\left (c + d x\right )^{\frac{7}{2}} \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{7 d^{5}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (5 a^{4} b d^{4} - 20 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 20 a b^{4} c^{3} d + 5 b^{5} c^{4}\right )}{5 d^{5}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}\right )}{3 d^{5}}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5*(d*x+c)**(1/2),x)

[Out]

2*(b**5*(c + d*x)**(13/2)/(13*d**5) + (c + d*x)**(11/2)*(5*a*b**4*d - 5*b**5*c)/(11*d**5) + (c + d*x)**(9/2)*(
10*a**2*b**3*d**2 - 20*a*b**4*c*d + 10*b**5*c**2)/(9*d**5) + (c + d*x)**(7/2)*(10*a**3*b**2*d**3 - 30*a**2*b**
3*c*d**2 + 30*a*b**4*c**2*d - 10*b**5*c**3)/(7*d**5) + (c + d*x)**(5/2)*(5*a**4*b*d**4 - 20*a**3*b**2*c*d**3 +
30*a**2*b**3*c**2*d**2 - 20*a*b**4*c**3*d + 5*b**5*c**4)/(5*d**5) + (c + d*x)**(3/2)*(a**5*d**5 - 5*a**4*b*c*
d**4 + 10*a**3*b**2*c**2*d**3 - 10*a**2*b**3*c**3*d**2 + 5*a*b**4*c**4*d - b**5*c**5)/(3*d**5))/d

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Giac [B]  time = 1.09713, size = 385, normalized size = 2.47 \begin{align*} \frac{2 \,{\left (3003 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{5} + \frac{3003 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a^{4} b}{d} + \frac{858 \,{\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^{3} b^{2}}{d^{2}} + \frac{286 \,{\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^{2} b^{3}}{d^{3}} + \frac{13 \,{\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 )} a b^{4}}{d^{4}} + \frac{{\left (693 \,{\left (d x + c\right )}^{\frac{13}{2}} - 4095 \,{\left (d x + c\right )}^{\frac{11}{2}} c + 10010 \,{\left (d x + c\right )}^{\frac{9}{2}} c^{2} - 12870 \,{\left (d x + c\right )}^{\frac{7}{2}} c^{3} + 9009 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{4} - 3003 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{5}\right )} b^{5}}{d^{5}}\right )}}{9009 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/9009*(3003*(d*x + c)^(3/2)*a^5 + 3003*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^4*b/d + 858*(15*(d*x + c)^
(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a^3*b^2/d^2 + 286*(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^2*b^3/d^3 + 13*(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)*a*b^4/d^4 + (6
93*(d*x + c)^(13/2) - 4095*(d*x + c)^(11/2)*c + 10010*(d*x + c)^(9/2)*c^2 - 12870*(d*x + c)^(7/2)*c^3 + 9009*(
d*x + c)^(5/2)*c^4 - 3003*(d*x + c)^(3/2)*c^5)*b^5/d^5)/d