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]

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

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Rubi [A]  time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.45, size = 99, normalized size = 1.39 \[ \frac {2 \, {\left (15 \, b^{2} d^{3} x^{3} + 8 \, b^{2} c^{3} - 28 \, a b c^{2} d + 35 \, a^{2} c d^{2} + 3 \, {\left (b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{2} - {\left (4 \, b^{2} c^{2} d - 14 \, a b c d^{2} - 35 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{105 \, d^{3}} \]

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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giac [B]  time = 1.29, size = 200, normalized size = 2.82 \[ \frac {2 \, {\left (105 \, \sqrt {d x + c} a^{2} c + 35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} + \frac {70 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a b c}{d} + \frac {7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} b^{2} c}{d^{2}} + \frac {14 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b}{d} + \frac {3 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{2}}{d^{2}}\right )}}{105 \, d} \]

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*(105*sqrt(d*x + c)*a^2*c + 35*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^2 + 70*((d*x + c)^(3/2) - 3*sqrt(d
*x + c)*c)*a*b*c/d + 7*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*b^2*c/d^2 + 14*(3*(d*
x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a*b/d + 3*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*
c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*b^2/d^2)/d

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maple [A]  time = 0.01, size = 63, normalized size = 0.89 \[ \frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (15 b^{2} x^{2} d^{2}+42 a b \,d^{2} x -12 b^{2} c d x +35 a^{2} d^{2}-28 a b c d +8 b^{2} c^{2}\right )}{105 d^{3}} \]

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 = 1.35, size = 68, normalized size = 0.96 \[ \frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} - 42 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 35 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{105 \, d^{3}} \]

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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mupad [B]  time = 0.24, size = 68, normalized size = 0.96 \[ \frac {2\,{\left (c+d\,x\right )}^{3/2}\,\left (15\,b^2\,{\left (c+d\,x\right )}^2+35\,a^2\,d^2+35\,b^2\,c^2-42\,b^2\,c\,\left (c+d\,x\right )+42\,a\,b\,d\,\left (c+d\,x\right )-70\,a\,b\,c\,d\right )}{105\,d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2*(c + d*x)^(1/2),x)

[Out]

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

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sympy [A]  time = 2.69, size = 85, normalized size = 1.20 \[ \frac {2 \left (\frac {b^{2} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{2}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (2 a b d - 2 b^{2} c\right )}{5 d^{2}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 d^{2}}\right )}{d} \]

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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