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 = 0.0229477, 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, Rules used = {43} \[ -\frac{4 b (c+d x)^{9/2} (b c-a d)}{9 d^3}+\frac{2 (c+d x)^{7/2} (b c-a d)^2}{7 d^3}+\frac{2 b^2 (c+d x)^{11/2}}{11 d^3} \]

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)

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

Mathematica [A]  time = 0.0553781, 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 = 0.006, size = 63, normalized size = 0.9 \begin{align*}{\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}}}} \end{align*}

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 = 0.952526, size = 92, normalized size = 1.3 \begin{align*} \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}} \end{align*}

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 [B]  time = 1.84682, size = 382, normalized size = 5.38 \begin{align*} \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}} \end{align*}

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 = 3.21486, size = 355, normalized size = 5. \begin{align*} \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} \end{align*}

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*sqrt(c + d*x)/(63*d**2) + 4*a*b*c**3*x*sqrt(c + d*x)/(63*d) + 2
0*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)/693 + 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 [B]  time = 1.09028, size = 491, normalized size = 6.92 \begin{align*} \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} + 33 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2}\right )} a^{2} + \frac{33 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2}\right )} b^{2} c^{2}}{d^{2}} + \frac{132 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2}\right )} a b c}{d} + \frac{22 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3}\right )} b^{2} c}{d^{2}} + \frac{22 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} - 135 \,{\left (d x + c\right )}^{\frac{7}{2}} c + 189 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{2} - 105 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{3}\right )} a b}{d} + \frac{{\left (315 \,{\left (d x + c\right )}^{\frac{11}{2}} - 1540 \,{\left (d x + c\right )}^{\frac{9}{2}} c + 2970 \,{\left (d x + c\right )}^{\frac{7}{2}} c^{2} - 2772 \,{\left (d x + c\right )}^{\frac{5}{2}} c^{3} + 1155 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{4}\right )} b^{2}}{d^{2}}\right )}}{3465 \, d} \end{align*}

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) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^
2)*a^2 + 33*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*b^2*c^2/d^2 + 132*(15*(d*x +
c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a*b*c/d + 22*(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)*b^2*c/d^2 + 22*(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/d + (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^2/d^2)/d