3.1401 \(\int (a+b x)^3 (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=100 \[ -\frac{6 b^2 (c+d x)^{11/2} (b c-a d)}{11 d^4}+\frac{2 b (c+d x)^{9/2} (b c-a d)^2}{3 d^4}-\frac{2 (c+d x)^{7/2} (b c-a d)^3}{7 d^4}+\frac{2 b^3 (c+d x)^{13/2}}{13 d^4} \]

[Out]

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

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Rubi [A]  time = 0.0312856, antiderivative size = 100, 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} \[ -\frac{6 b^2 (c+d x)^{11/2} (b c-a d)}{11 d^4}+\frac{2 b (c+d x)^{9/2} (b c-a d)^2}{3 d^4}-\frac{2 (c+d x)^{7/2} (b c-a d)^3}{7 d^4}+\frac{2 b^3 (c+d x)^{13/2}}{13 d^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(c + d*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.069674, size = 79, normalized size = 0.79 \[ \frac{2 (c+d x)^{7/2} \left (-819 b^2 (c+d x)^2 (b c-a d)+1001 b (c+d x) (b c-a d)^2-429 (b c-a d)^3+231 b^3 (c+d x)^3\right )}{3003 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2)*(-429*(b*c - a*d)^3 + 1001*b*(b*c - a*d)^2*(c + d*x) - 819*b^2*(b*c - a*d)*(c + d*x)^2 + 23
1*b^3*(c + d*x)^3))/(3003*d^4)

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Maple [A]  time = 0.005, size = 116, normalized size = 1.2 \begin{align*}{\frac{462\,{b}^{3}{x}^{3}{d}^{3}+1638\,a{b}^{2}{d}^{3}{x}^{2}-252\,{b}^{3}c{d}^{2}{x}^{2}+2002\,{a}^{2}b{d}^{3}x-728\,a{b}^{2}c{d}^{2}x+112\,{b}^{3}{c}^{2}dx+858\,{a}^{3}{d}^{3}-572\,{a}^{2}bc{d}^{2}+208\,a{b}^{2}{c}^{2}d-32\,{b}^{3}{c}^{3}}{3003\,{d}^{4}} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(d*x+c)^(5/2),x)

[Out]

2/3003*(d*x+c)^(7/2)*(231*b^3*d^3*x^3+819*a*b^2*d^3*x^2-126*b^3*c*d^2*x^2+1001*a^2*b*d^3*x-364*a*b^2*c*d^2*x+5
6*b^3*c^2*d*x+429*a^3*d^3-286*a^2*b*c*d^2+104*a*b^2*c^2*d-16*b^3*c^3)/d^4

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Maxima [A]  time = 0.965, size = 159, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (231 \,{\left (d x + c\right )}^{\frac{13}{2}} b^{3} - 819 \,{\left (b^{3} c - a b^{2} d\right )}{\left (d x + c\right )}^{\frac{11}{2}} + 1001 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )}{\left (d x + c\right )}^{\frac{9}{2}} - 429 \,{\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{7}{2}}\right )}}{3003 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/3003*(231*(d*x + c)^(13/2)*b^3 - 819*(b^3*c - a*b^2*d)*(d*x + c)^(11/2) + 1001*(b^3*c^2 - 2*a*b^2*c*d + a^2*
b*d^2)*(d*x + c)^(9/2) - 429*(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)^(7/2))/d^4

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Fricas [B]  time = 1.77024, size = 595, normalized size = 5.95 \begin{align*} \frac{2 \,{\left (231 \, b^{3} d^{6} x^{6} - 16 \, b^{3} c^{6} + 104 \, a b^{2} c^{5} d - 286 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} + 63 \,{\left (9 \, b^{3} c d^{5} + 13 \, a b^{2} d^{6}\right )} x^{5} + 7 \,{\left (53 \, b^{3} c^{2} d^{4} + 299 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} +{\left (5 \, b^{3} c^{3} d^{3} + 1469 \, a b^{2} c^{2} d^{4} + 2717 \, a^{2} b c d^{5} + 429 \, a^{3} d^{6}\right )} x^{3} - 3 \,{\left (2 \, b^{3} c^{4} d^{2} - 13 \, a b^{2} c^{3} d^{3} - 715 \, a^{2} b c^{2} d^{4} - 429 \, a^{3} c d^{5}\right )} x^{2} +{\left (8 \, b^{3} c^{5} d - 52 \, a b^{2} c^{4} d^{2} + 143 \, a^{2} b c^{3} d^{3} + 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt{d x + c}}{3003 \, d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

2/3003*(231*b^3*d^6*x^6 - 16*b^3*c^6 + 104*a*b^2*c^5*d - 286*a^2*b*c^4*d^2 + 429*a^3*c^3*d^3 + 63*(9*b^3*c*d^5
+ 13*a*b^2*d^6)*x^5 + 7*(53*b^3*c^2*d^4 + 299*a*b^2*c*d^5 + 143*a^2*b*d^6)*x^4 + (5*b^3*c^3*d^3 + 1469*a*b^2*
c^2*d^4 + 2717*a^2*b*c*d^5 + 429*a^3*d^6)*x^3 - 3*(2*b^3*c^4*d^2 - 13*a*b^2*c^3*d^3 - 715*a^2*b*c^2*d^4 - 429*
a^3*c*d^5)*x^2 + (8*b^3*c^5*d - 52*a*b^2*c^4*d^2 + 143*a^2*b*c^3*d^3 + 1287*a^3*c^2*d^4)*x)*sqrt(d*x + c)/d^4

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Sympy [A]  time = 4.37279, size = 549, normalized size = 5.49 \begin{align*} \begin{cases} \frac{2 a^{3} c^{3} \sqrt{c + d x}}{7 d} + \frac{6 a^{3} c^{2} x \sqrt{c + d x}}{7} + \frac{6 a^{3} c d x^{2} \sqrt{c + d x}}{7} + \frac{2 a^{3} d^{2} x^{3} \sqrt{c + d x}}{7} - \frac{4 a^{2} b c^{4} \sqrt{c + d x}}{21 d^{2}} + \frac{2 a^{2} b c^{3} x \sqrt{c + d x}}{21 d} + \frac{10 a^{2} b c^{2} x^{2} \sqrt{c + d x}}{7} + \frac{38 a^{2} b c d x^{3} \sqrt{c + d x}}{21} + \frac{2 a^{2} b d^{2} x^{4} \sqrt{c + d x}}{3} + \frac{16 a b^{2} c^{5} \sqrt{c + d x}}{231 d^{3}} - \frac{8 a b^{2} c^{4} x \sqrt{c + d x}}{231 d^{2}} + \frac{2 a b^{2} c^{3} x^{2} \sqrt{c + d x}}{77 d} + \frac{226 a b^{2} c^{2} x^{3} \sqrt{c + d x}}{231} + \frac{46 a b^{2} c d x^{4} \sqrt{c + d x}}{33} + \frac{6 a b^{2} d^{2} x^{5} \sqrt{c + d x}}{11} - \frac{32 b^{3} c^{6} \sqrt{c + d x}}{3003 d^{4}} + \frac{16 b^{3} c^{5} x \sqrt{c + d x}}{3003 d^{3}} - \frac{4 b^{3} c^{4} x^{2} \sqrt{c + d x}}{1001 d^{2}} + \frac{10 b^{3} c^{3} x^{3} \sqrt{c + d x}}{3003 d} + \frac{106 b^{3} c^{2} x^{4} \sqrt{c + d x}}{429} + \frac{54 b^{3} c d x^{5} \sqrt{c + d x}}{143} + \frac{2 b^{3} d^{2} x^{6} \sqrt{c + d x}}{13} & \text{for}\: d \neq 0 \\c^{\frac{5}{2}} \left (a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(d*x+c)**(5/2),x)

[Out]

Piecewise((2*a**3*c**3*sqrt(c + d*x)/(7*d) + 6*a**3*c**2*x*sqrt(c + d*x)/7 + 6*a**3*c*d*x**2*sqrt(c + d*x)/7 +
2*a**3*d**2*x**3*sqrt(c + d*x)/7 - 4*a**2*b*c**4*sqrt(c + d*x)/(21*d**2) + 2*a**2*b*c**3*x*sqrt(c + d*x)/(21*
d) + 10*a**2*b*c**2*x**2*sqrt(c + d*x)/7 + 38*a**2*b*c*d*x**3*sqrt(c + d*x)/21 + 2*a**2*b*d**2*x**4*sqrt(c + d
*x)/3 + 16*a*b**2*c**5*sqrt(c + d*x)/(231*d**3) - 8*a*b**2*c**4*x*sqrt(c + d*x)/(231*d**2) + 2*a*b**2*c**3*x**
2*sqrt(c + d*x)/(77*d) + 226*a*b**2*c**2*x**3*sqrt(c + d*x)/231 + 46*a*b**2*c*d*x**4*sqrt(c + d*x)/33 + 6*a*b*
*2*d**2*x**5*sqrt(c + d*x)/11 - 32*b**3*c**6*sqrt(c + d*x)/(3003*d**4) + 16*b**3*c**5*x*sqrt(c + d*x)/(3003*d*
*3) - 4*b**3*c**4*x**2*sqrt(c + d*x)/(1001*d**2) + 10*b**3*c**3*x**3*sqrt(c + d*x)/(3003*d) + 106*b**3*c**2*x*
*4*sqrt(c + d*x)/429 + 54*b**3*c*d*x**5*sqrt(c + d*x)/143 + 2*b**3*d**2*x**6*sqrt(c + d*x)/13, Ne(d, 0)), (c**
(5/2)*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), True))

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Giac [B]  time = 1.08548, size = 770, normalized size = 7.7 \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)^3*(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/45045*(15015*(d*x + c)^(3/2)*a^3*c^2 + 6006*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^3*c + 9009*(3*(d*x +
c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^2*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^3 + 1287*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a*b^2*c^2/d^2 + 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*c/d + 143*(35*(d*x + c)^(9/2) - 13
5*(d*x + c)^(7/2)*c + 189*(d*x + c)^(5/2)*c^2 - 105*(d*x + c)^(3/2)*c^3)*b^3*c^2/d^3 + 858*(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^2*c/d^2 + 429*(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/d + 26*(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^3*c/d^3 + 39*(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^2/d^2 + 5*(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^3/d^3)
/d