3.1402 \(\int (a+b x)^2 (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=71 \[ -\frac{4 b (c+d x)^{9/2} (b c-a d)}{9 d^3}+\frac{2 (c+d x)^{7/2} (b c-a d)^2}{7 d^3}+\frac{2 b^2 (c+d x)^{11/2}}{11 d^3} \]

[Out]

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

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Rubi [A]  time = 0.0675884, antiderivative size = 71, 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 \[ -\frac{4 b (c+d x)^{9/2} (b c-a d)}{9 d^3}+\frac{2 (c+d x)^{7/2} (b c-a d)^2}{7 d^3}+\frac{2 b^2 (c+d x)^{11/2}}{11 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.7428, size = 65, normalized size = 0.92 \[ \frac{2 b^{2} \left (c + d x\right )^{\frac{11}{2}}}{11 d^{3}} + \frac{4 b \left (c + d x\right )^{\frac{9}{2}} \left (a d - b c\right )}{9 d^{3}} + \frac{2 \left (c + d x\right )^{\frac{7}{2}} \left (a d - b c\right )^{2}}{7 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(d*x+c)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.0816965, size = 61, normalized size = 0.86 \[ \frac{2 (c+d x)^{7/2} \left (99 a^2 d^2+22 a b d (7 d x-2 c)+b^2 \left (8 c^2-28 c d x+63 d^2 x^2\right )\right )}{693 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(2*(c + d*x)^(7/2)*(99*a^2*d^2 + 22*a*b*d*(-2*c + 7*d*x) + b^2*(8*c^2 - 28*c*d*x
+ 63*d^2*x^2)))/(693*d^3)

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Maple [A]  time = 0.009, size = 63, normalized size = 0.9 \[{\frac{126\,{b}^{2}{x}^{2}{d}^{2}+308\,ab{d}^{2}x-56\,{b}^{2}cdx+198\,{a}^{2}{d}^{2}-88\,abcd+16\,{b}^{2}{c}^{2}}{693\,{d}^{3}} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(d*x+c)^(5/2),x)

[Out]

2/693*(d*x+c)^(7/2)*(63*b^2*d^2*x^2+154*a*b*d^2*x-28*b^2*c*d*x+99*a^2*d^2-44*a*b
*c*d+8*b^2*c^2)/d^3

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Maxima [A]  time = 1.3537, size = 92, normalized size = 1.3 \[ \frac{2 \,{\left (63 \,{\left (d x + c\right )}^{\frac{11}{2}} b^{2} - 154 \,{\left (b^{2} c - a b d\right )}{\left (d x + c\right )}^{\frac{9}{2}} + 99 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{7}{2}}\right )}}{693 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(63*(d*x + c)^(11/2)*b^2 - 154*(b^2*c - a*b*d)*(d*x + c)^(9/2) + 99*(b^2*c
^2 - 2*a*b*c*d + a^2*d^2)*(d*x + c)^(7/2))/d^3

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Fricas [A]  time = 0.219416, size = 235, normalized size = 3.31 \[ \frac{2 \,{\left (63 \, b^{2} d^{5} x^{5} + 8 \, b^{2} c^{5} - 44 \, a b c^{4} d + 99 \, a^{2} c^{3} d^{2} + 7 \,{\left (23 \, b^{2} c d^{4} + 22 \, a b d^{5}\right )} x^{4} +{\left (113 \, b^{2} c^{2} d^{3} + 418 \, a b c d^{4} + 99 \, a^{2} d^{5}\right )} x^{3} + 3 \,{\left (b^{2} c^{3} d^{2} + 110 \, a b c^{2} d^{3} + 99 \, a^{2} c d^{4}\right )} x^{2} -{\left (4 \, b^{2} c^{4} d - 22 \, a b c^{3} d^{2} - 297 \, a^{2} c^{2} d^{3}\right )} x\right )} \sqrt{d x + c}}{693 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/693*(63*b^2*d^5*x^5 + 8*b^2*c^5 - 44*a*b*c^4*d + 99*a^2*c^3*d^2 + 7*(23*b^2*c*
d^4 + 22*a*b*d^5)*x^4 + (113*b^2*c^2*d^3 + 418*a*b*c*d^4 + 99*a^2*d^5)*x^3 + 3*(
b^2*c^3*d^2 + 110*a*b*c^2*d^3 + 99*a^2*c*d^4)*x^2 - (4*b^2*c^4*d - 22*a*b*c^3*d^
2 - 297*a^2*c^2*d^3)*x)*sqrt(d*x + c)/d^3

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Sympy [A]  time = 4.74886, size = 355, normalized size = 5. \[ \begin{cases} \frac{2 a^{2} c^{3} \sqrt{c + d x}}{7 d} + \frac{6 a^{2} c^{2} x \sqrt{c + d x}}{7} + \frac{6 a^{2} c d x^{2} \sqrt{c + d x}}{7} + \frac{2 a^{2} d^{2} x^{3} \sqrt{c + d x}}{7} - \frac{8 a b c^{4} \sqrt{c + d x}}{63 d^{2}} + \frac{4 a b c^{3} x \sqrt{c + d x}}{63 d} + \frac{20 a b c^{2} x^{2} \sqrt{c + d x}}{21} + \frac{76 a b c d x^{3} \sqrt{c + d x}}{63} + \frac{4 a b d^{2} x^{4} \sqrt{c + d x}}{9} + \frac{16 b^{2} c^{5} \sqrt{c + d x}}{693 d^{3}} - \frac{8 b^{2} c^{4} x \sqrt{c + d x}}{693 d^{2}} + \frac{2 b^{2} c^{3} x^{2} \sqrt{c + d x}}{231 d} + \frac{226 b^{2} c^{2} x^{3} \sqrt{c + d x}}{693} + \frac{46 b^{2} c d x^{4} \sqrt{c + d x}}{99} + \frac{2 b^{2} d^{2} x^{5} \sqrt{c + d x}}{11} & \text{for}\: d \neq 0 \\c^{\frac{5}{2}} \left (a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2*(d*x+c)**(5/2),x)

[Out]

Piecewise((2*a**2*c**3*sqrt(c + d*x)/(7*d) + 6*a**2*c**2*x*sqrt(c + d*x)/7 + 6*a
**2*c*d*x**2*sqrt(c + d*x)/7 + 2*a**2*d**2*x**3*sqrt(c + d*x)/7 - 8*a*b*c**4*sqr
t(c + d*x)/(63*d**2) + 4*a*b*c**3*x*sqrt(c + d*x)/(63*d) + 20*a*b*c**2*x**2*sqrt
(c + d*x)/21 + 76*a*b*c*d*x**3*sqrt(c + d*x)/63 + 4*a*b*d**2*x**4*sqrt(c + d*x)/
9 + 16*b**2*c**5*sqrt(c + d*x)/(693*d**3) - 8*b**2*c**4*x*sqrt(c + d*x)/(693*d**
2) + 2*b**2*c**3*x**2*sqrt(c + d*x)/(231*d) + 226*b**2*c**2*x**3*sqrt(c + d*x)/6
93 + 46*b**2*c*d*x**4*sqrt(c + d*x)/99 + 2*b**2*d**2*x**5*sqrt(c + d*x)/11, Ne(d
, 0)), (c**(5/2)*(a**2*x + a*b*x**2 + b**2*x**3/3), True))

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GIAC/XCAS [A]  time = 0.22351, size = 585, normalized size = 8.24 \[ \frac{2 \,{\left (1155 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} c^{2} + 462 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a^{2} c + \frac{462 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a b c^{2}}{d} + \frac{33 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{12} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{12} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{12}\right )} b^{2} c^{2}}{d^{14}} + \frac{132 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{12} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{12} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{12}\right )} a b c}{d^{13}} + \frac{33 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{12} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{12} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{12}\right )} a^{2}}{d^{12}} + \frac{22 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} d^{24} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c d^{24} + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} d^{24} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3} d^{24}\right )} b^{2} c}{d^{26}} + \frac{22 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} d^{24} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c d^{24} + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} d^{24} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3} d^{24}\right )} a b}{d^{25}} + \frac{{\left (315 \,{\left (d x + c\right )}^{\frac{11}{2}} d^{40} - 1540 \,{\left (d x + c\right )}^{\frac{9}{2}} c d^{40} + 2970 \,{\left (d x + c\right )}^{\frac{7}{2}} c^{2} d^{40} - 2772 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{3} d^{40} + 1155 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{4} d^{40}\right )} b^{2}}{d^{42}}\right )}}{3465 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^2*(d*x + c)^(5/2),x, algorithm="giac")

[Out]

2/3465*(1155*(d*x + c)^(3/2)*a^2*c^2 + 462*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2
)*c)*a^2*c + 462*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a*b*c^2/d + 33*(15*(d
*x + c)^(7/2)*d^12 - 42*(d*x + c)^(5/2)*c*d^12 + 35*(d*x + c)^(3/2)*c^2*d^12)*b^
2*c^2/d^14 + 132*(15*(d*x + c)^(7/2)*d^12 - 42*(d*x + c)^(5/2)*c*d^12 + 35*(d*x
+ c)^(3/2)*c^2*d^12)*a*b*c/d^13 + 33*(15*(d*x + c)^(7/2)*d^12 - 42*(d*x + c)^(5/
2)*c*d^12 + 35*(d*x + c)^(3/2)*c^2*d^12)*a^2/d^12 + 22*(35*(d*x + c)^(9/2)*d^24
- 135*(d*x + c)^(7/2)*c*d^24 + 189*(d*x + c)^(5/2)*c^2*d^24 - 105*(d*x + c)^(3/2
)*c^3*d^24)*b^2*c/d^26 + 22*(35*(d*x + c)^(9/2)*d^24 - 135*(d*x + c)^(7/2)*c*d^2
4 + 189*(d*x + c)^(5/2)*c^2*d^24 - 105*(d*x + c)^(3/2)*c^3*d^24)*a*b/d^25 + (315
*(d*x + c)^(11/2)*d^40 - 1540*(d*x + c)^(9/2)*c*d^40 + 2970*(d*x + c)^(7/2)*c^2*
d^40 - 2772*(d*x + c)^(5/2)*c^3*d^40 + 1155*(d*x + c)^(3/2)*c^4*d^40)*b^2/d^42)/
d