3.1378 \(\int (a+b x)^2 \sqrt{c+d x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0247449, 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} \[ -\frac{4 b (c+d x)^{5/2} (b c-a d)}{5 d^3}+\frac{2 (c+d x)^{3/2} (b c-a d)^2}{3 d^3}+\frac{2 b^2 (c+d x)^{7/2}}{7 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*Sqrt[c + d*x],x]

[Out]

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

Mathematica [A]  time = 0.0353577, size = 61, normalized size = 0.86 \[ \frac{2 (c+d x)^{3/2} \left (35 a^2 d^2+14 a b d (3 d x-2 c)+b^2 \left (8 c^2-12 c d x+15 d^2 x^2\right )\right )}{105 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + d*x)^(3/2)*(35*a^2*d^2 + 14*a*b*d*(-2*c + 3*d*x) + b^2*(8*c^2 - 12*c*d*x + 15*d^2*x^2)))/(105*d^3)

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Maple [A]  time = 0.004, size = 63, normalized size = 0.9 \begin{align*}{\frac{30\,{b}^{2}{x}^{2}{d}^{2}+84\,ab{d}^{2}x-24\,{b}^{2}cdx+70\,{a}^{2}{d}^{2}-56\,abcd+16\,{b}^{2}{c}^{2}}{105\,{d}^{3}} \left ( dx+c \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(d*x+c)^(1/2),x)

[Out]

2/105*(d*x+c)^(3/2)*(15*b^2*d^2*x^2+42*a*b*d^2*x-12*b^2*c*d*x+35*a^2*d^2-28*a*b*c*d+8*b^2*c^2)/d^3

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Maxima [A]  time = 0.978978, size = 92, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} b^{2} - 42 \,{\left (b^{2} c - a b d\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 35 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\left (d x + c\right )}^{\frac{3}{2}}\right )}}{105 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.9933, size = 220, normalized size = 3.1 \begin{align*} \frac{2 \,{\left (15 \, b^{2} d^{3} x^{3} + 8 \, b^{2} c^{3} - 28 \, a b c^{2} d + 35 \, a^{2} c d^{2} + 3 \,{\left (b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{2} -{\left (4 \, b^{2} c^{2} d - 14 \, a b c d^{2} - 35 \, a^{2} d^{3}\right )} x\right )} \sqrt{d x + c}}{105 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*b^2*d^3*x^3 + 8*b^2*c^3 - 28*a*b*c^2*d + 35*a^2*c*d^2 + 3*(b^2*c*d^2 + 14*a*b*d^3)*x^2 - (4*b^2*c^2*
d - 14*a*b*c*d^2 - 35*a^2*d^3)*x)*sqrt(d*x + c)/d^3

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Sympy [A]  time = 1.89997, size = 85, normalized size = 1.2 \begin{align*} \frac{2 \left (\frac{b^{2} \left (c + d x\right )^{\frac{7}{2}}}{7 d^{2}} + \frac{\left (c + d x\right )^{\frac{5}{2}} \left (2 a b d - 2 b^{2} c\right )}{5 d^{2}} + \frac{\left (c + d x\right )^{\frac{3}{2}} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 d^{2}}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(d*x+c)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.06055, size = 113, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{3}{2}} a^{2} + \frac{14 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} a b}{d} + \frac{{\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}}{d^{2}}\right )}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/105*(35*(d*x + c)^(3/2)*a^2 + 14*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*a*b/d + (15*(d*x + c)^(7/2) - 42*
(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2)*b^2/d^2)/d