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

Optimal. Leaf size=19 \[ \frac {2}{5} d \left (a+b x+c x^2\right )^{5/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {629} \begin {gather*} \frac {2}{5} d \left (a+b x+c x^2\right )^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(a + b*x + c*x^2)^(5/2))/5

Rule 629

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.95 \begin {gather*} \frac {2}{5} d (a+x (b+c x))^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(a + x*(b + c*x))^(5/2))/5

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IntegrateAlgebraic [A]  time = 0.02, size = 19, normalized size = 1.00 \begin {gather*} \frac {2}{5} d \left (a+b x+c x^2\right )^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(a + b*x + c*x^2)^(5/2))/5

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fricas [B]  time = 0.42, size = 55, normalized size = 2.89 \begin {gather*} \frac {2}{5} \, {\left (c^{2} d x^{4} + 2 \, b c d x^{3} + 2 \, a b d x + {\left (b^{2} + 2 \, a c\right )} d x^{2} + a^{2} d\right )} \sqrt {c x^{2} + b x + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.21, size = 15, normalized size = 0.79 \begin {gather*} \frac {2}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c*x^2 + b*x + a)^(5/2)*d

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maple [A]  time = 0.04, size = 16, normalized size = 0.84 \begin {gather*} \frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} d}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.35, size = 15, normalized size = 0.79 \begin {gather*} \frac {2}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(c*x^2 + b*x + a)^(5/2)*d

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mupad [B]  time = 0.55, size = 15, normalized size = 0.79 \begin {gather*} \frac {2\,d\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*d*(a + b*x + c*x^2)^(5/2))/5

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sympy [B]  time = 0.71, size = 146, normalized size = 7.68 \begin {gather*} \frac {2 a^{2} d \sqrt {a + b x + c x^{2}}}{5} + \frac {4 a b d x \sqrt {a + b x + c x^{2}}}{5} + \frac {4 a c d x^{2} \sqrt {a + b x + c x^{2}}}{5} + \frac {2 b^{2} d x^{2} \sqrt {a + b x + c x^{2}}}{5} + \frac {4 b c d x^{3} \sqrt {a + b x + c x^{2}}}{5} + \frac {2 c^{2} d x^{4} \sqrt {a + b x + c x^{2}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*d*sqrt(a + b*x + c*x**2)/5 + 4*a*b*d*x*sqrt(a + b*x + c*x**2)/5 + 4*a*c*d*x**2*sqrt(a + b*x + c*x**2)/5
+ 2*b**2*d*x**2*sqrt(a + b*x + c*x**2)/5 + 4*b*c*d*x**3*sqrt(a + b*x + c*x**2)/5 + 2*c**2*d*x**4*sqrt(a + b*x
+ c*x**2)/5

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