3.1413 \(\int \frac{(a+b x)^5}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=154 \[ -\frac{10 b^4 (c+d x)^{9/2} (b c-a d)}{9 d^6}+\frac{20 b^3 (c+d x)^{7/2} (b c-a d)^2}{7 d^6}-\frac{4 b^2 (c+d x)^{5/2} (b c-a d)^3}{d^6}+\frac{10 b (c+d x)^{3/2} (b c-a d)^4}{3 d^6}-\frac{2 \sqrt{c+d x} (b c-a d)^5}{d^6}+\frac{2 b^5 (c+d x)^{11/2}}{11 d^6} \]

[Out]

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

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Rubi [A]  time = 0.0507118, antiderivative size = 154, 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} \[ -\frac{10 b^4 (c+d x)^{9/2} (b c-a d)}{9 d^6}+\frac{20 b^3 (c+d x)^{7/2} (b c-a d)^2}{7 d^6}-\frac{4 b^2 (c+d x)^{5/2} (b c-a d)^3}{d^6}+\frac{10 b (c+d x)^{3/2} (b c-a d)^4}{3 d^6}-\frac{2 \sqrt{c+d x} (b c-a d)^5}{d^6}+\frac{2 b^5 (c+d x)^{11/2}}{11 d^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/Sqrt[c + d*x],x]

[Out]

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

Mathematica [A]  time = 0.0822461, size = 123, normalized size = 0.8 \[ \frac{2 \sqrt{c+d x} \left (-1386 b^2 (c+d x)^2 (b c-a d)^3+990 b^3 (c+d x)^3 (b c-a d)^2-385 b^4 (c+d x)^4 (b c-a d)+1155 b (c+d x) (b c-a d)^4-693 (b c-a d)^5+63 b^5 (c+d x)^5\right )}{693 d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c + d*x]*(-693*(b*c - a*d)^5 + 1155*b*(b*c - a*d)^4*(c + d*x) - 1386*b^2*(b*c - a*d)^3*(c + d*x)^2 + 9
90*b^3*(b*c - a*d)^2*(c + d*x)^3 - 385*b^4*(b*c - a*d)*(c + d*x)^4 + 63*b^5*(c + d*x)^5))/(693*d^6)

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Maple [B]  time = 0.006, size = 273, normalized size = 1.8 \begin{align*}{\frac{126\,{b}^{5}{x}^{5}{d}^{5}+770\,a{b}^{4}{d}^{5}{x}^{4}-140\,{b}^{5}c{d}^{4}{x}^{4}+1980\,{a}^{2}{b}^{3}{d}^{5}{x}^{3}-880\,a{b}^{4}c{d}^{4}{x}^{3}+160\,{b}^{5}{c}^{2}{d}^{3}{x}^{3}+2772\,{a}^{3}{b}^{2}{d}^{5}{x}^{2}-2376\,{a}^{2}{b}^{3}c{d}^{4}{x}^{2}+1056\,a{b}^{4}{c}^{2}{d}^{3}{x}^{2}-192\,{b}^{5}{c}^{3}{d}^{2}{x}^{2}+2310\,{a}^{4}b{d}^{5}x-3696\,{a}^{3}{b}^{2}c{d}^{4}x+3168\,{a}^{2}{b}^{3}{c}^{2}{d}^{3}x-1408\,a{b}^{4}{c}^{3}{d}^{2}x+256\,{b}^{5}{c}^{4}dx+1386\,{a}^{5}{d}^{5}-4620\,{a}^{4}bc{d}^{4}+7392\,{a}^{3}{b}^{2}{c}^{2}{d}^{3}-6336\,{a}^{2}{b}^{3}{c}^{3}{d}^{2}+2816\,a{b}^{4}{c}^{4}d-512\,{b}^{5}{c}^{5}}{693\,{d}^{6}}\sqrt{dx+c}} \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]

2/693*(d*x+c)^(1/2)*(63*b^5*d^5*x^5+385*a*b^4*d^5*x^4-70*b^5*c*d^4*x^4+990*a^2*b^3*d^5*x^3-440*a*b^4*c*d^4*x^3
+80*b^5*c^2*d^3*x^3+1386*a^3*b^2*d^5*x^2-1188*a^2*b^3*c*d^4*x^2+528*a*b^4*c^2*d^3*x^2-96*b^5*c^3*d^2*x^2+1155*
a^4*b*d^5*x-1848*a^3*b^2*c*d^4*x+1584*a^2*b^3*c^2*d^3*x-704*a*b^4*c^3*d^2*x+128*b^5*c^4*d*x+693*a^5*d^5-2310*a
^4*b*c*d^4+3696*a^3*b^2*c^2*d^3-3168*a^2*b^3*c^3*d^2+1408*a*b^4*c^4*d-256*b^5*c^5)/d^6

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Maxima [B]  time = 0.983722, size = 382, normalized size = 2.48 \begin{align*} \frac{2 \,{\left (693 \, \sqrt{d x + c} a^{5} + \frac{1155 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} a^{4} b}{d} + \frac{462 \,{\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^{3} b^{2}}{d^{2}} + \frac{198 \,{\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^{2} b^{3}}{d^{3}} + \frac{11 \,{\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 )} a b^{4}}{d^{4}} + \frac{{\left (63 \,{\left (d x + c\right )}^{\frac{11}{2}} - 385 \,{\left (d x + c\right )}^{\frac{9}{2}} c + 990 \,{\left (d x + c\right )}^{\frac{7}{2}} c^{2} - 1386 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{3} + 1155 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{4} - 693 \, \sqrt{d x + c} c^{5}\right )} b^{5}}{d^{5}}\right )}}{693 \, 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="maxima")

[Out]

2/693*(693*sqrt(d*x + c)*a^5 + 1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*b/d + 462*(3*(d*x + c)^(5/2) - 1
0*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b^2/d^2 + 198*(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^2*b^3/d^3 + 11*(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)*a*b^4/d^4 + (63*(d*x + c)^(11/2) - 385*
(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d
*x + c)*c^5)*b^5/d^5)/d

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Fricas [A]  time = 1.82477, size = 586, normalized size = 3.81 \begin{align*} \frac{2 \,{\left (63 \, b^{5} d^{5} x^{5} - 256 \, b^{5} c^{5} + 1408 \, a b^{4} c^{4} d - 3168 \, a^{2} b^{3} c^{3} d^{2} + 3696 \, a^{3} b^{2} c^{2} d^{3} - 2310 \, a^{4} b c d^{4} + 693 \, a^{5} d^{5} - 35 \,{\left (2 \, b^{5} c d^{4} - 11 \, a b^{4} d^{5}\right )} x^{4} + 10 \,{\left (8 \, b^{5} c^{2} d^{3} - 44 \, a b^{4} c d^{4} + 99 \, a^{2} b^{3} d^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} c^{3} d^{2} - 88 \, a b^{4} c^{2} d^{3} + 198 \, a^{2} b^{3} c d^{4} - 231 \, a^{3} b^{2} d^{5}\right )} x^{2} +{\left (128 \, b^{5} c^{4} d - 704 \, a b^{4} c^{3} d^{2} + 1584 \, a^{2} b^{3} c^{2} d^{3} - 1848 \, a^{3} b^{2} c d^{4} + 1155 \, a^{4} b d^{5}\right )} x\right )} \sqrt{d x + c}}{693 \, 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]

2/693*(63*b^5*d^5*x^5 - 256*b^5*c^5 + 1408*a*b^4*c^4*d - 3168*a^2*b^3*c^3*d^2 + 3696*a^3*b^2*c^2*d^3 - 2310*a^
4*b*c*d^4 + 693*a^5*d^5 - 35*(2*b^5*c*d^4 - 11*a*b^4*d^5)*x^4 + 10*(8*b^5*c^2*d^3 - 44*a*b^4*c*d^4 + 99*a^2*b^
3*d^5)*x^3 - 6*(16*b^5*c^3*d^2 - 88*a*b^4*c^2*d^3 + 198*a^2*b^3*c*d^4 - 231*a^3*b^2*d^5)*x^2 + (128*b^5*c^4*d
- 704*a*b^4*c^3*d^2 + 1584*a^2*b^3*c^2*d^3 - 1848*a^3*b^2*c*d^4 + 1155*a^4*b*d^5)*x)*sqrt(d*x + c)/d^6

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Sympy [A]  time = 55.8994, size = 728, normalized size = 4.73 \begin{align*} \text{result too large to display} \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]

Piecewise((-(2*a**5*c/sqrt(c + d*x) + 2*a**5*(-c/sqrt(c + d*x) - sqrt(c + d*x)) + 10*a**4*b*c*(-c/sqrt(c + d*x
) - sqrt(c + d*x))/d + 10*a**4*b*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d + 20*a**3*b**
2*c*(c**2/sqrt(c + d*x) + 2*c*sqrt(c + d*x) - (c + d*x)**(3/2)/3)/d**2 + 20*a**3*b**2*(-c**3/sqrt(c + d*x) - 3
*c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**2 + 20*a**2*b**3*c*(-c**3/sqrt(c + d*x) - 3*
c**2*sqrt(c + d*x) + c*(c + d*x)**(3/2) - (c + d*x)**(5/2)/5)/d**3 + 20*a**2*b**3*(c**4/sqrt(c + d*x) + 4*c**3
*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*x)**(5/2)/5 - (c + d*x)**(7/2)/7)/d**3 + 10*a*b**4*c*(c*
*4/sqrt(c + d*x) + 4*c**3*sqrt(c + d*x) - 2*c**2*(c + d*x)**(3/2) + 4*c*(c + d*x)**(5/2)/5 - (c + d*x)**(7/2)/
7)/d**4 + 10*a*b**4*(-c**5/sqrt(c + d*x) - 5*c**4*sqrt(c + d*x) + 10*c**3*(c + d*x)**(3/2)/3 - 2*c**2*(c + d*x
)**(5/2) + 5*c*(c + d*x)**(7/2)/7 - (c + d*x)**(9/2)/9)/d**4 + 2*b**5*c*(-c**5/sqrt(c + d*x) - 5*c**4*sqrt(c +
d*x) + 10*c**3*(c + d*x)**(3/2)/3 - 2*c**2*(c + d*x)**(5/2) + 5*c*(c + d*x)**(7/2)/7 - (c + d*x)**(9/2)/9)/d*
*5 + 2*b**5*(c**6/sqrt(c + d*x) + 6*c**5*sqrt(c + d*x) - 5*c**4*(c + d*x)**(3/2) + 4*c**3*(c + d*x)**(5/2) - 1
5*c**2*(c + d*x)**(7/2)/7 + 2*c*(c + d*x)**(9/2)/3 - (c + d*x)**(11/2)/11)/d**5)/d, Ne(d, 0)), (Piecewise((a**
5*x, Eq(b, 0)), ((a + b*x)**6/(6*b), True))/sqrt(c), True))

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Giac [B]  time = 1.0831, size = 382, normalized size = 2.48 \begin{align*} \frac{2 \,{\left (693 \, \sqrt{d x + c} a^{5} + \frac{1155 \,{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} a^{4} b}{d} + \frac{462 \,{\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^{3} b^{2}}{d^{2}} + \frac{198 \,{\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^{2} b^{3}}{d^{3}} + \frac{11 \,{\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 )} a b^{4}}{d^{4}} + \frac{{\left (63 \,{\left (d x + c\right )}^{\frac{11}{2}} - 385 \,{\left (d x + c\right )}^{\frac{9}{2}} c + 990 \,{\left (d x + c\right )}^{\frac{7}{2}} c^{2} - 1386 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{3} + 1155 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{4} - 693 \, \sqrt{d x + c} c^{5}\right )} b^{5}}{d^{5}}\right )}}{693 \, 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]

2/693*(693*sqrt(d*x + c)*a^5 + 1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^4*b/d + 462*(3*(d*x + c)^(5/2) - 1
0*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a^3*b^2/d^2 + 198*(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^2*b^3/d^3 + 11*(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)*a*b^4/d^4 + (63*(d*x + c)^(11/2) - 385*
(d*x + c)^(9/2)*c + 990*(d*x + c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693*sqrt(d
*x + c)*c^5)*b^5/d^5)/d