3.1390 \(\int (a+b x)^2 (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=71 \[ -\frac{4 b (c+d x)^{7/2} (b c-a d)}{7 d^3}+\frac{2 (c+d x)^{5/2} (b c-a d)^2}{5 d^3}+\frac{2 b^2 (c+d x)^{9/2}}{9 d^3} \]

[Out]

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

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Rubi [A]  time = 0.0237218, 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} \[ -\frac{4 b (c+d x)^{7/2} (b c-a d)}{7 d^3}+\frac{2 (c+d x)^{5/2} (b c-a d)^2}{5 d^3}+\frac{2 b^2 (c+d x)^{9/2}}{9 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(c + d*x)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0411658, size = 61, normalized size = 0.86 \[ \frac{2 (c+d x)^{5/2} \left (63 a^2 d^2+18 a b d (5 d x-2 c)+b^2 \left (8 c^2-20 c d x+35 d^2 x^2\right )\right )}{315 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(5/2)*(63*a^2*d^2 + 18*a*b*d*(-2*c + 5*d*x) + b^2*(8*c^2 - 20*c*d*x + 35*d^2*x^2)))/(315*d^3)

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Maple [A]  time = 0.005, size = 63, normalized size = 0.9 \begin{align*}{\frac{70\,{b}^{2}{x}^{2}{d}^{2}+180\,ab{d}^{2}x-40\,{b}^{2}cdx+126\,{a}^{2}{d}^{2}-72\,abcd+16\,{b}^{2}{c}^{2}}{315\,{d}^{3}} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(d*x+c)^(5/2)*(35*b^2*d^2*x^2+90*a*b*d^2*x-20*b^2*c*d*x+63*a^2*d^2-36*a*b*c*d+8*b^2*c^2)/d^3

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Maxima [A]  time = 0.973623, size = 92, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{9}{2}} b^{2} - 90 \,{\left (b^{2} c - a b d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 63 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{5}{2}}\right )}}{315 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(d*x + c)^(9/2)*b^2 - 90*(b^2*c - a*b*d)*(d*x + c)^(7/2) + 63*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(d*x +
c)^(5/2))/d^3

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Fricas [B]  time = 1.95459, size = 300, normalized size = 4.23 \begin{align*} \frac{2 \,{\left (35 \, b^{2} d^{4} x^{4} + 8 \, b^{2} c^{4} - 36 \, a b c^{3} d + 63 \, a^{2} c^{2} d^{2} + 10 \,{\left (5 \, b^{2} c d^{3} + 9 \, a b d^{4}\right )} x^{3} + 3 \,{\left (b^{2} c^{2} d^{2} + 48 \, a b c d^{3} + 21 \, a^{2} d^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} c^{3} d - 9 \, a b c^{2} d^{2} - 63 \, a^{2} c d^{3}\right )} x\right )} \sqrt{d x + c}}{315 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*b^2*d^4*x^4 + 8*b^2*c^4 - 36*a*b*c^3*d + 63*a^2*c^2*d^2 + 10*(5*b^2*c*d^3 + 9*a*b*d^4)*x^3 + 3*(b^2*
c^2*d^2 + 48*a*b*c*d^3 + 21*a^2*d^4)*x^2 - 2*(2*b^2*c^3*d - 9*a*b*c^2*d^2 - 63*a^2*c*d^3)*x)*sqrt(d*x + c)/d^3

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Sympy [A]  time = 7.67253, size = 240, normalized size = 3.38 \begin{align*} a^{2} c \left (\begin{cases} \sqrt{c} x & \text{for}\: d = 0 \\\frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{2} \left (- \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d} + \frac{4 a b c \left (- \frac{c \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{\left (c + d x\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} + \frac{4 a b \left (\frac{c^{2} \left (c + d x\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + d x\right )^{\frac{5}{2}}}{5} + \frac{\left (c + d x\right )^{\frac{7}{2}}}{7}\right )}{d^{2}} + \frac{2 b^{2} c \left (\frac{c^{2} \left (c + d x\right )^{\frac{3}{2}}}{3} - \frac{2 c \left (c + d x\right )^{\frac{5}{2}}}{5} + \frac{\left (c + d x\right )^{\frac{7}{2}}}{7}\right )}{d^{3}} + \frac{2 b^{2} \left (- \frac{c^{3} \left (c + d x\right )^{\frac{3}{2}}}{3} + \frac{3 c^{2} \left (c + d x\right )^{\frac{5}{2}}}{5} - \frac{3 c \left (c + d x\right )^{\frac{7}{2}}}{7} + \frac{\left (c + d x\right )^{\frac{9}{2}}}{9}\right )}{d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.06499, size = 274, normalized size = 3.86 \begin{align*} \frac{2 \,{\left (105 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} c + 21 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a^{2} + \frac{42 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a b c}{d} + \frac{3 \,{\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}{d^{2}} + \frac{6 \,{\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}{d} + \frac{{\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}}{d^{2}}\right )}}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(105*(d*x + c)^(3/2)*a^2*c + 21*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a^2 + 42*(3*(d*x + c)^(5/2) -
5*(d*x + c)^(3/2)*c)*a*b*c/d + 3*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*b^2*c/d^
2 + 6*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*a*b/d + (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/d^2)/d