3.1400 \(\int (a+b x)^4 (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=129 \[ -\frac{8 b^3 (c+d x)^{13/2} (b c-a d)}{13 d^5}+\frac{12 b^2 (c+d x)^{11/2} (b c-a d)^2}{11 d^5}-\frac{8 b (c+d x)^{9/2} (b c-a d)^3}{9 d^5}+\frac{2 (c+d x)^{7/2} (b c-a d)^4}{7 d^5}+\frac{2 b^4 (c+d x)^{15/2}}{15 d^5} \]

[Out]

(2*(b*c - a*d)^4*(c + d*x)^(7/2))/(7*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(9/2))/(9*d^5) + (12*b^2*(b*c - a*d)^
2*(c + d*x)^(11/2))/(11*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(13/2))/(13*d^5) + (2*b^4*(c + d*x)^(15/2))/(15*d^
5)

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Rubi [A]  time = 0.0427892, 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} \[ -\frac{8 b^3 (c+d x)^{13/2} (b c-a d)}{13 d^5}+\frac{12 b^2 (c+d x)^{11/2} (b c-a d)^2}{11 d^5}-\frac{8 b (c+d x)^{9/2} (b c-a d)^3}{9 d^5}+\frac{2 (c+d x)^{7/2} (b c-a d)^4}{7 d^5}+\frac{2 b^4 (c+d x)^{15/2}}{15 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)^4*(c + d*x)^(7/2))/(7*d^5) - (8*b*(b*c - a*d)^3*(c + d*x)^(9/2))/(9*d^5) + (12*b^2*(b*c - a*d)^
2*(c + d*x)^(11/2))/(11*d^5) - (8*b^3*(b*c - a*d)*(c + d*x)^(13/2))/(13*d^5) + (2*b^4*(c + d*x)^(15/2))/(15*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 (c+d x)^{5/2} \, dx &=\int \left (\frac{(-b c+a d)^4 (c+d x)^{5/2}}{d^4}-\frac{4 b (b c-a d)^3 (c+d x)^{7/2}}{d^4}+\frac{6 b^2 (b c-a d)^2 (c+d x)^{9/2}}{d^4}-\frac{4 b^3 (b c-a d) (c+d x)^{11/2}}{d^4}+\frac{b^4 (c+d x)^{13/2}}{d^4}\right ) \, dx\\ &=\frac{2 (b c-a d)^4 (c+d x)^{7/2}}{7 d^5}-\frac{8 b (b c-a d)^3 (c+d x)^{9/2}}{9 d^5}+\frac{12 b^2 (b c-a d)^2 (c+d x)^{11/2}}{11 d^5}-\frac{8 b^3 (b c-a d) (c+d x)^{13/2}}{13 d^5}+\frac{2 b^4 (c+d x)^{15/2}}{15 d^5}\\ \end{align*}

Mathematica [A]  time = 0.127086, size = 101, normalized size = 0.78 \[ \frac{2 (c+d x)^{7/2} \left (24570 b^2 (c+d x)^2 (b c-a d)^2-13860 b^3 (c+d x)^3 (b c-a d)-20020 b (c+d x) (b c-a d)^3+6435 (b c-a d)^4+3003 b^4 (c+d x)^4\right )}{45045 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2)*(6435*(b*c - a*d)^4 - 20020*b*(b*c - a*d)^3*(c + d*x) + 24570*b^2*(b*c - a*d)^2*(c + d*x)^2
- 13860*b^3*(b*c - a*d)*(c + d*x)^3 + 3003*b^4*(c + d*x)^4))/(45045*d^5)

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Maple [A]  time = 0.005, size = 186, normalized size = 1.4 \begin{align*}{\frac{6006\,{b}^{4}{x}^{4}{d}^{4}+27720\,a{b}^{3}{d}^{4}{x}^{3}-3696\,{b}^{4}c{d}^{3}{x}^{3}+49140\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-15120\,a{b}^{3}c{d}^{3}{x}^{2}+2016\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+40040\,{a}^{3}b{d}^{4}x-21840\,{a}^{2}{b}^{2}c{d}^{3}x+6720\,a{b}^{3}{c}^{2}{d}^{2}x-896\,{b}^{4}{c}^{3}dx+12870\,{a}^{4}{d}^{4}-11440\,{a}^{3}bc{d}^{3}+6240\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-1920\,a{b}^{3}{c}^{3}d+256\,{b}^{4}{c}^{4}}{45045\,{d}^{5}} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(d*x+c)^(5/2),x)

[Out]

2/45045*(d*x+c)^(7/2)*(3003*b^4*d^4*x^4+13860*a*b^3*d^4*x^3-1848*b^4*c*d^3*x^3+24570*a^2*b^2*d^4*x^2-7560*a*b^
3*c*d^3*x^2+1008*b^4*c^2*d^2*x^2+20020*a^3*b*d^4*x-10920*a^2*b^2*c*d^3*x+3360*a*b^3*c^2*d^2*x-448*b^4*c^3*d*x+
6435*a^4*d^4-5720*a^3*b*c*d^3+3120*a^2*b^2*c^2*d^2-960*a*b^3*c^3*d+128*b^4*c^4)/d^5

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Maxima [A]  time = 0.967286, size = 244, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (3003 \,{\left (d x + c\right )}^{\frac{15}{2}} b^{4} - 13860 \,{\left (b^{4} c - a b^{3} d\right )}{\left (d x + c\right )}^{\frac{13}{2}} + 24570 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{11}{2}} - 20020 \,{\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{9}{2}} + 6435 \,{\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{7}{2}}\right )}}{45045 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*(d*x + c)^(15/2)*b^4 - 13860*(b^4*c - a*b^3*d)*(d*x + c)^(13/2) + 24570*(b^4*c^2 - 2*a*b^3*c*d +
a^2*b^2*d^2)*(d*x + c)^(11/2) - 20020*(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)^(9/2)
+ 6435*(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)^(7/2))/d^5

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Fricas [B]  time = 1.94254, size = 855, normalized size = 6.63 \begin{align*} \frac{2 \,{\left (3003 \, b^{4} d^{7} x^{7} + 128 \, b^{4} c^{7} - 960 \, a b^{3} c^{6} d + 3120 \, a^{2} b^{2} c^{5} d^{2} - 5720 \, a^{3} b c^{4} d^{3} + 6435 \, a^{4} c^{3} d^{4} + 231 \,{\left (31 \, b^{4} c d^{6} + 60 \, a b^{3} d^{7}\right )} x^{6} + 63 \,{\left (71 \, b^{4} c^{2} d^{5} + 540 \, a b^{3} c d^{6} + 390 \, a^{2} b^{2} d^{7}\right )} x^{5} + 35 \,{\left (b^{4} c^{3} d^{4} + 636 \, a b^{3} c^{2} d^{5} + 1794 \, a^{2} b^{2} c d^{6} + 572 \, a^{3} b d^{7}\right )} x^{4} - 5 \,{\left (8 \, b^{4} c^{4} d^{3} - 60 \, a b^{3} c^{3} d^{4} - 8814 \, a^{2} b^{2} c^{2} d^{5} - 10868 \, a^{3} b c d^{6} - 1287 \, a^{4} d^{7}\right )} x^{3} + 3 \,{\left (16 \, b^{4} c^{5} d^{2} - 120 \, a b^{3} c^{4} d^{3} + 390 \, a^{2} b^{2} c^{3} d^{4} + 14300 \, a^{3} b c^{2} d^{5} + 6435 \, a^{4} c d^{6}\right )} x^{2} -{\left (64 \, b^{4} c^{6} d - 480 \, a b^{3} c^{5} d^{2} + 1560 \, a^{2} b^{2} c^{4} d^{3} - 2860 \, a^{3} b c^{3} d^{4} - 19305 \, a^{4} c^{2} d^{5}\right )} x\right )} \sqrt{d x + c}}{45045 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

2/45045*(3003*b^4*d^7*x^7 + 128*b^4*c^7 - 960*a*b^3*c^6*d + 3120*a^2*b^2*c^5*d^2 - 5720*a^3*b*c^4*d^3 + 6435*a
^4*c^3*d^4 + 231*(31*b^4*c*d^6 + 60*a*b^3*d^7)*x^6 + 63*(71*b^4*c^2*d^5 + 540*a*b^3*c*d^6 + 390*a^2*b^2*d^7)*x
^5 + 35*(b^4*c^3*d^4 + 636*a*b^3*c^2*d^5 + 1794*a^2*b^2*c*d^6 + 572*a^3*b*d^7)*x^4 - 5*(8*b^4*c^4*d^3 - 60*a*b
^3*c^3*d^4 - 8814*a^2*b^2*c^2*d^5 - 10868*a^3*b*c*d^6 - 1287*a^4*d^7)*x^3 + 3*(16*b^4*c^5*d^2 - 120*a*b^3*c^4*
d^3 + 390*a^2*b^2*c^3*d^4 + 14300*a^3*b*c^2*d^5 + 6435*a^4*c*d^6)*x^2 - (64*b^4*c^6*d - 480*a*b^3*c^5*d^2 + 15
60*a^2*b^2*c^4*d^3 - 2860*a^3*b*c^3*d^4 - 19305*a^4*c^2*d^5)*x)*sqrt(d*x + c)/d^5

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Sympy [A]  time = 25.1616, size = 960, normalized size = 7.44 \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)**4*(d*x+c)**(5/2),x)

[Out]

a**4*c**2*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 4*a**4*c*(-c*(c + d*x)**(3/2)/3
+ (c + d*x)**(5/2)/5)/d + 2*a**4*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d +
8*a**3*b*c**2*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 16*a**3*b*c*(c**2*(c + d*x)**(3/2)/3 - 2*c*(
c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**2 + 8*a**3*b*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5
- 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**2 + 12*a**2*b**2*c**2*(c**2*(c + d*x)**(3/2)/3 - 2*c*(c + d*
x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**3 + 24*a**2*b**2*c*(-c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5
- 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**3 + 12*a**2*b**2*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)
**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2)/9 + (c + d*x)**(11/2)/11)/d**3 + 8*a*b**3*c**2*(-
c**3*(c + d*x)**(3/2)/3 + 3*c**2*(c + d*x)**(5/2)/5 - 3*c*(c + d*x)**(7/2)/7 + (c + d*x)**(9/2)/9)/d**4 + 16*a
*b**3*c*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2
)/9 + (c + d*x)**(11/2)/11)/d**4 + 8*a*b**3*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 10*c**3*(c + d
*x)**(7/2)/7 + 10*c**2*(c + d*x)**(9/2)/9 - 5*c*(c + d*x)**(11/2)/11 + (c + d*x)**(13/2)/13)/d**4 + 2*b**4*c**
2*(c**4*(c + d*x)**(3/2)/3 - 4*c**3*(c + d*x)**(5/2)/5 + 6*c**2*(c + d*x)**(7/2)/7 - 4*c*(c + d*x)**(9/2)/9 +
(c + d*x)**(11/2)/11)/d**5 + 4*b**4*c*(-c**5*(c + d*x)**(3/2)/3 + c**4*(c + d*x)**(5/2) - 10*c**3*(c + d*x)**(
7/2)/7 + 10*c**2*(c + d*x)**(9/2)/9 - 5*c*(c + d*x)**(11/2)/11 + (c + d*x)**(13/2)/13)/d**5 + 2*b**4*(c**6*(c
+ d*x)**(3/2)/3 - 6*c**5*(c + d*x)**(5/2)/5 + 15*c**4*(c + d*x)**(7/2)/7 - 20*c**3*(c + d*x)**(9/2)/9 + 15*c**
2*(c + d*x)**(11/2)/11 - 6*c*(c + d*x)**(13/2)/13 + (c + d*x)**(15/2)/15)/d**5

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Giac [B]  time = 1.1228, size = 1094, normalized size = 8.48 \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)^4*(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/45045*(15015*(d*x + c)^(3/2)*a^4*c^2 + 6006*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^4*c + 12012*(3*(d*x
+ c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^3*b*c^2/d + 429*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^
(3/2)*c^2)*a^4 + 2574*(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*c^2/d^2 + 3
432*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a^3*b*c/d + 572*(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*c^2/d^3 + 1716*(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^2*c/d^2 + 572*(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^3*b/d + 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)*b^4*c^2/d^4 + 104*(315*(d*x + c)^(11/2) - 1540*(d*x + c)^(9/2)*c + 2970*(d*x + c)^(7/2)*c^2 - 277
2*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4)*a*b^3*c/d^3 + 78*(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^2*b^2/d^2 + 10*(693*(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^4*c/d^4 + 20*(693*(d*x + c)^(13/2) - 4095*(d*x + c)^(11/2)*c + 100
10*(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)*a*b^
3/d^3 + (3003*(d*x + c)^(15/2) - 20790*(d*x + c)^(13/2)*c + 61425*(d*x + c)^(11/2)*c^2 - 100100*(d*x + c)^(9/2
)*c^3 + 96525*(d*x + c)^(7/2)*c^4 - 54054*(d*x + c)^(5/2)*c^5 + 15015*(d*x + c)^(3/2)*c^6)*b^4/d^4)/d