3.1376 \(\int (a+b x)^4 \sqrt{c+d x} \, dx\)

Optimal. Leaf size=129 \[ -\frac{8 b^3 (c+d x)^{9/2} (b c-a d)}{9 d^5}+\frac{12 b^2 (c+d x)^{7/2} (b c-a d)^2}{7 d^5}-\frac{8 b (c+d x)^{5/2} (b c-a d)^3}{5 d^5}+\frac{2 (c+d x)^{3/2} (b c-a d)^4}{3 d^5}+\frac{2 b^4 (c+d x)^{11/2}}{11 d^5} \]

[Out]

(2*(b*c - a*d)^4*(c + d*x)^(3/2))/(3*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(5/2))/(5*d^5) + (12*b^2*(b*c - a*d)^
2*(c + d*x)^(7/2))/(7*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(9/2))/(9*d^5) + (2*b^4*(c + d*x)^(11/2))/(11*d^5)

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Rubi [A]  time = 0.0519554, antiderivative size = 129, 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} \[ -\frac{8 b^3 (c+d x)^{9/2} (b c-a d)}{9 d^5}+\frac{12 b^2 (c+d x)^{7/2} (b c-a d)^2}{7 d^5}-\frac{8 b (c+d x)^{5/2} (b c-a d)^3}{5 d^5}+\frac{2 (c+d x)^{3/2} (b c-a d)^4}{3 d^5}+\frac{2 b^4 (c+d x)^{11/2}}{11 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*Sqrt[c + d*x],x]

[Out]

(2*(b*c - a*d)^4*(c + d*x)^(3/2))/(3*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(5/2))/(5*d^5) + (12*b^2*(b*c - a*d)^
2*(c + d*x)^(7/2))/(7*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(9/2))/(9*d^5) + (2*b^4*(c + d*x)^(11/2))/(11*d^5)

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)^4 \sqrt{c+d x} \, dx &=\int \left (\frac{(-b c+a d)^4 \sqrt{c+d x}}{d^4}-\frac{4 b (b c-a d)^3 (c+d x)^{3/2}}{d^4}+\frac{6 b^2 (b c-a d)^2 (c+d x)^{5/2}}{d^4}-\frac{4 b^3 (b c-a d) (c+d x)^{7/2}}{d^4}+\frac{b^4 (c+d x)^{9/2}}{d^4}\right ) \, dx\\ &=\frac{2 (b c-a d)^4 (c+d x)^{3/2}}{3 d^5}-\frac{8 b (b c-a d)^3 (c+d x)^{5/2}}{5 d^5}+\frac{12 b^2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^5}-\frac{8 b^3 (b c-a d) (c+d x)^{9/2}}{9 d^5}+\frac{2 b^4 (c+d x)^{11/2}}{11 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0867557, size = 101, normalized size = 0.78 \[ \frac{2 (c+d x)^{3/2} \left (2970 b^2 (c+d x)^2 (b c-a d)^2-1540 b^3 (c+d x)^3 (b c-a d)-2772 b (c+d x) (b c-a d)^3+1155 (b c-a d)^4+315 b^4 (c+d x)^4\right )}{3465 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2)*(1155*(b*c - a*d)^4 - 2772*b*(b*c - a*d)^3*(c + d*x) + 2970*b^2*(b*c - a*d)^2*(c + d*x)^2 -
1540*b^3*(b*c - a*d)*(c + d*x)^3 + 315*b^4*(c + d*x)^4))/(3465*d^5)

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Maple [A]  time = 0.005, size = 186, normalized size = 1.4 \begin{align*}{\frac{630\,{b}^{4}{x}^{4}{d}^{4}+3080\,a{b}^{3}{d}^{4}{x}^{3}-560\,{b}^{4}c{d}^{3}{x}^{3}+5940\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-2640\,a{b}^{3}c{d}^{3}{x}^{2}+480\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+5544\,{a}^{3}b{d}^{4}x-4752\,{a}^{2}{b}^{2}c{d}^{3}x+2112\,a{b}^{3}{c}^{2}{d}^{2}x-384\,{b}^{4}{c}^{3}dx+2310\,{a}^{4}{d}^{4}-3696\,{a}^{3}bc{d}^{3}+3168\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-1408\,a{b}^{3}{c}^{3}d+256\,{b}^{4}{c}^{4}}{3465\,{d}^{5}} \left ( dx+c \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(d*x+c)^(1/2),x)

[Out]

2/3465*(d*x+c)^(3/2)*(315*b^4*d^4*x^4+1540*a*b^3*d^4*x^3-280*b^4*c*d^3*x^3+2970*a^2*b^2*d^4*x^2-1320*a*b^3*c*d
^3*x^2+240*b^4*c^2*d^2*x^2+2772*a^3*b*d^4*x-2376*a^2*b^2*c*d^3*x+1056*a*b^3*c^2*d^2*x-192*b^4*c^3*d*x+1155*a^4
*d^4-1848*a^3*b*c*d^3+1584*a^2*b^2*c^2*d^2-704*a*b^3*c^3*d+128*b^4*c^4)/d^5

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Maxima [A]  time = 0.956078, size = 244, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (315 \,{\left (d x + c\right )}^{\frac{11}{2}} b^{4} - 1540 \,{\left (b^{4} c - a b^{3} d\right )}{\left (d x + c\right )}^{\frac{9}{2}} + 2970 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{7}{2}} - 2772 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 1155 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\left (d x + c\right )}^{\frac{3}{2}}\right )}}{3465 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(d*x + c)^(11/2)*b^4 - 1540*(b^4*c - a*b^3*d)*(d*x + c)^(9/2) + 2970*(b^4*c^2 - 2*a*b^3*c*d + a^2*
b^2*d^2)*(d*x + c)^(7/2) - 2772*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*(d*x + c)^(5/2) + 1155
*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*(d*x + c)^(3/2))/d^5

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Fricas [B]  time = 1.95661, size = 547, normalized size = 4.24 \begin{align*} \frac{2 \,{\left (315 \, b^{4} d^{5} x^{5} + 128 \, b^{4} c^{5} - 704 \, a b^{3} c^{4} d + 1584 \, a^{2} b^{2} c^{3} d^{2} - 1848 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} + 35 \,{\left (b^{4} c d^{4} + 44 \, a b^{3} d^{5}\right )} x^{4} - 10 \,{\left (4 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} - 297 \, a^{2} b^{2} d^{5}\right )} x^{3} + 6 \,{\left (8 \, b^{4} c^{3} d^{2} - 44 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} + 462 \, a^{3} b d^{5}\right )} x^{2} -{\left (64 \, b^{4} c^{4} d - 352 \, a b^{3} c^{3} d^{2} + 792 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} - 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt{d x + c}}{3465 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*b^4*d^5*x^5 + 128*b^4*c^5 - 704*a*b^3*c^4*d + 1584*a^2*b^2*c^3*d^2 - 1848*a^3*b*c^2*d^3 + 1155*a^4
*c*d^4 + 35*(b^4*c*d^4 + 44*a*b^3*d^5)*x^4 - 10*(4*b^4*c^2*d^3 - 22*a*b^3*c*d^4 - 297*a^2*b^2*d^5)*x^3 + 6*(8*
b^4*c^3*d^2 - 44*a*b^3*c^2*d^3 + 99*a^2*b^2*c*d^4 + 462*a^3*b*d^5)*x^2 - (64*b^4*c^4*d - 352*a*b^3*c^3*d^2 + 7
92*a^2*b^2*c^2*d^3 - 924*a^3*b*c*d^4 - 1155*a^4*d^5)*x)*sqrt(d*x + c)/d^5

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Sympy [A]  time = 3.06919, size = 223, normalized size = 1.73 \begin{align*} \frac{2 \left (\frac{b^{4} \left (c + d x\right )^{\frac{11}{2}}}{11 d^{4}} + \frac{\left (c + d x\right )^{\frac{9}{2}} \left (4 a b^{3} d - 4 b^{4} c\right )}{9 d^{4}} + \frac{\left (c + d x\right )^{\frac{7}{2}} \left (6 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 6 b^{4} c^{2}\right )}{7 d^{4}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (4 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 4 b^{4} c^{3}\right )}{5 d^{4}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{3 d^{4}}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4*(d*x+c)**(1/2),x)

[Out]

2*(b**4*(c + d*x)**(11/2)/(11*d**4) + (c + d*x)**(9/2)*(4*a*b**3*d - 4*b**4*c)/(9*d**4) + (c + d*x)**(7/2)*(6*
a**2*b**2*d**2 - 12*a*b**3*c*d + 6*b**4*c**2)/(7*d**4) + (c + d*x)**(5/2)*(4*a**3*b*d**3 - 12*a**2*b**2*c*d**2
+ 12*a*b**3*c**2*d - 4*b**4*c**3)/(5*d**4) + (c + d*x)**(3/2)*(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2
*d**2 - 4*a*b**3*c**3*d + b**4*c**4)/(3*d**4))/d

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Giac [A]  time = 1.09752, size = 278, normalized size = 2.16 \begin{align*} \frac{2 \,{\left (1155 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{4} + \frac{924 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a^{3} b}{d} + \frac{198 \,{\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^{2}}{d^{2}} + \frac{44 \,{\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^{3}}{d^{3}} + \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^{4}}{d^{4}}\right )}}{3465 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/3465*(1155*(d*x + c)^(3/2)*a^4 + 924*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^3*b/d + 198*(15*(d*x + c)^(
7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a^2*b^2/d^2 + 44*(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^3/d^3 + (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^4/d^4)/d