3.1388 \(\int (a+b x)^4 (c+d x)^{3/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0419296, 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} \[ -\frac{8 b^3 (c+d x)^{11/2} (b c-a d)}{11 d^5}+\frac{4 b^2 (c+d x)^{9/2} (b c-a d)^2}{3 d^5}-\frac{8 b (c+d x)^{7/2} (b c-a d)^3}{7 d^5}+\frac{2 (c+d x)^{5/2} (b c-a d)^4}{5 d^5}+\frac{2 b^4 (c+d x)^{13/2}}{13 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4*(c + d*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0953262, size = 101, normalized size = 0.78 \[ \frac{2 (c+d x)^{5/2} \left (10010 b^2 (c+d x)^2 (b c-a d)^2-5460 b^3 (c+d x)^3 (b c-a d)-8580 b (c+d x) (b c-a d)^3+3003 (b c-a d)^4+1155 b^4 (c+d x)^4\right )}{15015 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4*(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2)*(3003*(b*c - a*d)^4 - 8580*b*(b*c - a*d)^3*(c + d*x) + 10010*b^2*(b*c - a*d)^2*(c + d*x)^2
- 5460*b^3*(b*c - a*d)*(c + d*x)^3 + 1155*b^4*(c + d*x)^4))/(15015*d^5)

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Maple [A]  time = 0.006, size = 186, normalized size = 1.4 \begin{align*}{\frac{2310\,{b}^{4}{x}^{4}{d}^{4}+10920\,a{b}^{3}{d}^{4}{x}^{3}-1680\,{b}^{4}c{d}^{3}{x}^{3}+20020\,{a}^{2}{b}^{2}{d}^{4}{x}^{2}-7280\,a{b}^{3}c{d}^{3}{x}^{2}+1120\,{b}^{4}{c}^{2}{d}^{2}{x}^{2}+17160\,{a}^{3}b{d}^{4}x-11440\,{a}^{2}{b}^{2}c{d}^{3}x+4160\,a{b}^{3}{c}^{2}{d}^{2}x-640\,{b}^{4}{c}^{3}dx+6006\,{a}^{4}{d}^{4}-6864\,{a}^{3}bc{d}^{3}+4576\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-1664\,a{b}^{3}{c}^{3}d+256\,{b}^{4}{c}^{4}}{15015\,{d}^{5}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4*(d*x+c)^(3/2),x)

[Out]

2/15015*(d*x+c)^(5/2)*(1155*b^4*d^4*x^4+5460*a*b^3*d^4*x^3-840*b^4*c*d^3*x^3+10010*a^2*b^2*d^4*x^2-3640*a*b^3*
c*d^3*x^2+560*b^4*c^2*d^2*x^2+8580*a^3*b*d^4*x-5720*a^2*b^2*c*d^3*x+2080*a*b^3*c^2*d^2*x-320*b^4*c^3*d*x+3003*
a^4*d^4-3432*a^3*b*c*d^3+2288*a^2*b^2*c^2*d^2-832*a*b^3*c^3*d+128*b^4*c^4)/d^5

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Maxima [A]  time = 0.977237, size = 244, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (1155 \,{\left (d x + c\right )}^{\frac{13}{2}} b^{4} - 5460 \,{\left (b^{4} c - a b^{3} d\right )}{\left (d x + c\right )}^{\frac{11}{2}} + 10010 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{9}{2}} - 8580 \,{\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{7}{2}} + 3003 \,{\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{5}{2}}\right )}}{15015 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(3/2),x, algorithm="maxima")

[Out]

2/15015*(1155*(d*x + c)^(13/2)*b^4 - 5460*(b^4*c - a*b^3*d)*(d*x + c)^(11/2) + 10010*(b^4*c^2 - 2*a*b^3*c*d +
a^2*b^2*d^2)*(d*x + c)^(9/2) - 8580*(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)^(7/2) +
3003*(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)^(5/2))/d^5

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Fricas [B]  time = 2.13225, size = 702, normalized size = 5.44 \begin{align*} \frac{2 \,{\left (1155 \, b^{4} d^{6} x^{6} + 128 \, b^{4} c^{6} - 832 \, a b^{3} c^{5} d + 2288 \, a^{2} b^{2} c^{4} d^{2} - 3432 \, a^{3} b c^{3} d^{3} + 3003 \, a^{4} c^{2} d^{4} + 210 \,{\left (7 \, b^{4} c d^{5} + 26 \, a b^{3} d^{6}\right )} x^{5} + 35 \,{\left (b^{4} c^{2} d^{4} + 208 \, a b^{3} c d^{5} + 286 \, a^{2} b^{2} d^{6}\right )} x^{4} - 20 \,{\left (2 \, b^{4} c^{3} d^{3} - 13 \, a b^{3} c^{2} d^{4} - 715 \, a^{2} b^{2} c d^{5} - 429 \, a^{3} b d^{6}\right )} x^{3} + 3 \,{\left (16 \, b^{4} c^{4} d^{2} - 104 \, a b^{3} c^{3} d^{3} + 286 \, a^{2} b^{2} c^{2} d^{4} + 4576 \, a^{3} b c d^{5} + 1001 \, a^{4} d^{6}\right )} x^{2} - 2 \,{\left (32 \, b^{4} c^{5} d - 208 \, a b^{3} c^{4} d^{2} + 572 \, a^{2} b^{2} c^{3} d^{3} - 858 \, a^{3} b c^{2} d^{4} - 3003 \, a^{4} c d^{5}\right )} x\right )} \sqrt{d x + c}}{15015 \, d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4*(d*x+c)^(3/2),x, algorithm="fricas")

[Out]

2/15015*(1155*b^4*d^6*x^6 + 128*b^4*c^6 - 832*a*b^3*c^5*d + 2288*a^2*b^2*c^4*d^2 - 3432*a^3*b*c^3*d^3 + 3003*a
^4*c^2*d^4 + 210*(7*b^4*c*d^5 + 26*a*b^3*d^6)*x^5 + 35*(b^4*c^2*d^4 + 208*a*b^3*c*d^5 + 286*a^2*b^2*d^6)*x^4 -
20*(2*b^4*c^3*d^3 - 13*a*b^3*c^2*d^4 - 715*a^2*b^2*c*d^5 - 429*a^3*b*d^6)*x^3 + 3*(16*b^4*c^4*d^2 - 104*a*b^3
*c^3*d^3 + 286*a^2*b^2*c^2*d^4 + 4576*a^3*b*c*d^5 + 1001*a^4*d^6)*x^2 - 2*(32*b^4*c^5*d - 208*a*b^3*c^4*d^2 +
572*a^2*b^2*c^3*d^3 - 858*a^3*b*c^2*d^4 - 3003*a^4*c*d^5)*x)*sqrt(d*x + c)/d^5

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Sympy [A]  time = 15.5252, size = 559, normalized size = 4.33 \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)**(3/2),x)

[Out]

a**4*c*Piecewise((sqrt(c)*x, Eq(d, 0)), (2*(c + d*x)**(3/2)/(3*d), True)) + 2*a**4*(-c*(c + d*x)**(3/2)/3 + (c
+ d*x)**(5/2)/5)/d + 8*a**3*b*c*(-c*(c + d*x)**(3/2)/3 + (c + d*x)**(5/2)/5)/d**2 + 8*a**3*b*(c**2*(c + d*x)*
*(3/2)/3 - 2*c*(c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**2 + 12*a**2*b**2*c*(c**2*(c + d*x)**(3/2)/3 - 2*c*(
c + d*x)**(5/2)/5 + (c + d*x)**(7/2)/7)/d**3 + 12*a**2*b**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**3 + 8*a*b**3*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**4 + 8*a*b**3*(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 + 2*b**4*
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**5 + 2*b**4*(-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

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Giac [B]  time = 1.08234, size = 640, normalized size = 4.96 \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)^(3/2),x, algorithm="giac")

[Out]

2/45045*(15015*(d*x + c)^(3/2)*a^4*c + 3003*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^4 + 12012*(3*(d*x + c)
^(5/2) - 5*(d*x + c)^(3/2)*c)*a^3*b*c/d + 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/d^2 + 1716*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a^3*b/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/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^2*b^2/
d^2 + 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/d^4 + 52*(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^3/d^3 + 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^4/d^4)/d