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

Optimal. Leaf size=55 \[ \frac {(b d+2 c d x)^{11/2}}{44 c^2 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{28 c^2 d} \]

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Rubi [A]  time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {683} \begin {gather*} \frac {(b d+2 c d x)^{11/2}}{44 c^2 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{28 c^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(7/2))/(28*c^2*d) + (b*d + 2*c*d*x)^(11/2)/(44*c^2*d^3)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {align*} \int (b d+2 c d x)^{5/2} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (-b^2+4 a c\right ) (b d+2 c d x)^{5/2}}{4 c}+\frac {(b d+2 c d x)^{9/2}}{4 c d^2}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{28 c^2 d}+\frac {(b d+2 c d x)^{11/2}}{44 c^2 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 45, normalized size = 0.82 \begin {gather*} \frac {\left (c \left (11 a+7 c x^2\right )-b^2+7 b c x\right ) (d (b+2 c x))^{7/2}}{77 c^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d*(b + 2*c*x))^(7/2)*(-b^2 + 7*b*c*x + c*(11*a + 7*c*x^2)))/(77*c^2*d)

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IntegrateAlgebraic [A]  time = 0.08, size = 46, normalized size = 0.84 \begin {gather*} \frac {\left (11 a c-b^2+7 b c x+7 c^2 x^2\right ) (b d+2 c d x)^{7/2}}{77 c^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*d + 2*c*d*x)^(7/2)*(-b^2 + 11*a*c + 7*b*c*x + 7*c^2*x^2))/(77*c^2*d)

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fricas [B]  time = 0.40, size = 122, normalized size = 2.22 \begin {gather*} \frac {{\left (56 \, c^{5} d^{2} x^{5} + 140 \, b c^{4} d^{2} x^{4} + 2 \, {\left (59 \, b^{2} c^{3} + 44 \, a c^{4}\right )} d^{2} x^{3} + {\left (37 \, b^{3} c^{2} + 132 \, a b c^{3}\right )} d^{2} x^{2} + {\left (b^{4} c + 66 \, a b^{2} c^{2}\right )} d^{2} x - {\left (b^{5} - 11 \, a b^{3} c\right )} d^{2}\right )} \sqrt {2 \, c d x + b d}}{77 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/77*(56*c^5*d^2*x^5 + 140*b*c^4*d^2*x^4 + 2*(59*b^2*c^3 + 44*a*c^4)*d^2*x^3 + (37*b^3*c^2 + 132*a*b*c^3)*d^2*
x^2 + (b^4*c + 66*a*b^2*c^2)*d^2*x - (b^5 - 11*a*b^3*c)*d^2)*sqrt(2*c*d*x + b*d)/c^2

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giac [B]  time = 0.19, size = 567, normalized size = 10.31 \begin {gather*} \frac {13860 \, \sqrt {2 \, c d x + b d} a b^{3} d^{2} - 13860 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a b^{2} d - \frac {2310 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} b^{4} d}{c} + 2772 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} a b + \frac {1617 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b^{3}}{c} - \frac {396 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} a}{d} - \frac {891 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )} b^{2}}{c d} + \frac {55 \, {\left (315 \, \sqrt {2 \, c d x + b d} b^{4} d^{4} - 420 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{3} d^{3} + 378 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{2} - 180 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b d + 35 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}\right )} b}{c d^{2}} - \frac {5 \, {\left (693 \, \sqrt {2 \, c d x + b d} b^{5} d^{5} - 1155 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} d^{4} + 1386 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{3} d^{3} - 990 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{2} d^{2} + 385 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} b d - 63 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}}\right )}}{c d^{3}}}{13860 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13860*(13860*sqrt(2*c*d*x + b*d)*a*b^3*d^2 - 13860*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*a*b^2
*d - 2310*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*b^4*d/c + 2772*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 -
10*(2*c*d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*a*b + 1617*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*
d*x + b*d)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*b^3/c - 396*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*
d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*a/d - 891*(35*sqrt(2*c*d*x + b*d)*b
^3*d^3 - 35*(2*c*d*x + b*d)^(3/2)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))*b^2/(c*d)
+ 55*(315*sqrt(2*c*d*x + b*d)*b^4*d^4 - 420*(2*c*d*x + b*d)^(3/2)*b^3*d^3 + 378*(2*c*d*x + b*d)^(5/2)*b^2*d^2
- 180*(2*c*d*x + b*d)^(7/2)*b*d + 35*(2*c*d*x + b*d)^(9/2))*b/(c*d^2) - 5*(693*sqrt(2*c*d*x + b*d)*b^5*d^5 - 1
155*(2*c*d*x + b*d)^(3/2)*b^4*d^4 + 1386*(2*c*d*x + b*d)^(5/2)*b^3*d^3 - 990*(2*c*d*x + b*d)^(7/2)*b^2*d^2 + 3
85*(2*c*d*x + b*d)^(9/2)*b*d - 63*(2*c*d*x + b*d)^(11/2))/(c*d^3))/c

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maple [A]  time = 0.04, size = 46, normalized size = 0.84 \begin {gather*} \frac {\left (2 c x +b \right ) \left (7 c^{2} x^{2}+7 b c x +11 a c -b^{2}\right ) \left (2 c d x +b d \right )^{\frac {5}{2}}}{77 c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(5/2)*(c*x^2+b*x+a),x)

[Out]

1/77*(2*c*x+b)*(7*c^2*x^2+7*b*c*x+11*a*c-b^2)*(2*c*d*x+b*d)^(5/2)/c^2

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maxima [A]  time = 1.35, size = 46, normalized size = 0.84 \begin {gather*} -\frac {11 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 7 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}}}{308 \, c^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/308*(11*(2*c*d*x + b*d)^(7/2)*(b^2 - 4*a*c)*d^2 - 7*(2*c*d*x + b*d)^(11/2))/(c^2*d^3)

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mupad [B]  time = 0.06, size = 39, normalized size = 0.71 \begin {gather*} \frac {{\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,\left (44\,a\,c+7\,{\left (b+2\,c\,x\right )}^2-11\,b^2\right )}{308\,c^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2),x)

[Out]

((b*d + 2*c*d*x)^(7/2)*(44*a*c + 7*(b + 2*c*x)^2 - 11*b^2))/(308*c^2*d)

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sympy [A]  time = 3.85, size = 289, normalized size = 5.25 \begin {gather*} \begin {cases} \frac {a b^{3} d^{2} \sqrt {b d + 2 c d x}}{7 c} + \frac {6 a b^{2} d^{2} x \sqrt {b d + 2 c d x}}{7} + \frac {12 a b c d^{2} x^{2} \sqrt {b d + 2 c d x}}{7} + \frac {8 a c^{2} d^{2} x^{3} \sqrt {b d + 2 c d x}}{7} - \frac {b^{5} d^{2} \sqrt {b d + 2 c d x}}{77 c^{2}} + \frac {b^{4} d^{2} x \sqrt {b d + 2 c d x}}{77 c} + \frac {37 b^{3} d^{2} x^{2} \sqrt {b d + 2 c d x}}{77} + \frac {118 b^{2} c d^{2} x^{3} \sqrt {b d + 2 c d x}}{77} + \frac {20 b c^{2} d^{2} x^{4} \sqrt {b d + 2 c d x}}{11} + \frac {8 c^{3} d^{2} x^{5} \sqrt {b d + 2 c d x}}{11} & \text {for}\: c \neq 0 \\\left (b d\right )^{\frac {5}{2}} \left (a x + \frac {b x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*b**3*d**2*sqrt(b*d + 2*c*d*x)/(7*c) + 6*a*b**2*d**2*x*sqrt(b*d + 2*c*d*x)/7 + 12*a*b*c*d**2*x**2*
sqrt(b*d + 2*c*d*x)/7 + 8*a*c**2*d**2*x**3*sqrt(b*d + 2*c*d*x)/7 - b**5*d**2*sqrt(b*d + 2*c*d*x)/(77*c**2) + b
**4*d**2*x*sqrt(b*d + 2*c*d*x)/(77*c) + 37*b**3*d**2*x**2*sqrt(b*d + 2*c*d*x)/77 + 118*b**2*c*d**2*x**3*sqrt(b
*d + 2*c*d*x)/77 + 20*b*c**2*d**2*x**4*sqrt(b*d + 2*c*d*x)/11 + 8*c**3*d**2*x**5*sqrt(b*d + 2*c*d*x)/11, Ne(c,
0)), ((b*d)**(5/2)*(a*x + b*x**2/2), True))

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