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

Optimal. Leaf size=23 \[ \frac {2 (d (a+b x)+c)^{5/2}}{5 b d} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \begin {gather*} \frac {2 (d (a+b x)+c)^{5/2}}{5 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*(a + b*x))^(3/2),x]

[Out]

(2*(c + d*(a + b*x))^(5/2))/(5*b*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (d (a+b x)+c)^{5/2}}{5 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*(a + b*x))^(3/2),x]

[Out]

(2*(c + d*(a + b*x))^(5/2))/(5*b*d)

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IntegrateAlgebraic [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (a d+b d x+c)^{5/2}}{5 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(c + a*d + b*d*x)^(5/2))/(5*b*d)

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fricas [B]  time = 0.87, size = 59, normalized size = 2.57 \begin {gather*} \frac {2 \, {\left (b^{2} d^{2} x^{2} + a^{2} d^{2} + 2 \, a c d + c^{2} + 2 \, {\left (a b d^{2} + b c d\right )} x\right )} \sqrt {b d x + a d + c}}{5 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 1.04, size = 195, normalized size = 8.48 \begin {gather*} \frac {2 \, {\left (30 \, \sqrt {b d x + a d + c} a^{2} d^{2} - 10 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} a d + 60 \, \sqrt {b d x + a d + c} a c d - 10 \, {\left (3 \, \sqrt {b d x + a d + c} a d - {\left (b d x + a d + c\right )}^{\frac {3}{2}} + 3 \, \sqrt {b d x + a d + c} c\right )} a d + 3 \, {\left (b d x + a d + c\right )}^{\frac {5}{2}} - 10 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} c + 30 \, \sqrt {b d x + a d + c} c^{2} - 10 \, {\left (3 \, \sqrt {b d x + a d + c} a d - {\left (b d x + a d + c\right )}^{\frac {3}{2}} + 3 \, \sqrt {b d x + a d + c} c\right )} c\right )}}{15 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(30*sqrt(b*d*x + a*d + c)*a^2*d^2 - 10*(b*d*x + a*d + c)^(3/2)*a*d + 60*sqrt(b*d*x + a*d + c)*a*c*d - 10*
(3*sqrt(b*d*x + a*d + c)*a*d - (b*d*x + a*d + c)^(3/2) + 3*sqrt(b*d*x + a*d + c)*c)*a*d + 3*(b*d*x + a*d + c)^
(5/2) - 10*(b*d*x + a*d + c)^(3/2)*c + 30*sqrt(b*d*x + a*d + c)*c^2 - 10*(3*sqrt(b*d*x + a*d + c)*a*d - (b*d*x
+ a*d + c)^(3/2) + 3*sqrt(b*d*x + a*d + c)*c)*c)/(b*d)

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maple [A]  time = 0.00, size = 20, normalized size = 0.87 \begin {gather*} \frac {2 \left (b d x +a d +c \right )^{\frac {5}{2}}}{5 b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.88, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {5}{2}}}{5 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*((b*x + a)*d + c)^(5/2)/(b*d)

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mupad [B]  time = 0.17, size = 45, normalized size = 1.96 \begin {gather*} \sqrt {c+d\,\left (a+b\,x\right )}\,\left (x\,\left (\frac {4\,c}{5}+\frac {4\,a\,d}{5}\right )+\frac {2\,{\left (c+a\,d\right )}^2}{5\,b\,d}+\frac {2\,b\,d\,x^2}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*(a + b*x))^(3/2),x)

[Out]

(c + d*(a + b*x))^(1/2)*(x*((4*c)/5 + (4*a*d)/5) + (2*(c + a*d)^2)/(5*b*d) + (2*b*d*x^2)/5)

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sympy [A]  time = 5.27, size = 156, normalized size = 6.78 \begin {gather*} \begin {cases} c^{\frac {3}{2}} x & \text {for}\: b = 0 \wedge d = 0 \\x \left (a d + c\right )^{\frac {3}{2}} & \text {for}\: b = 0 \\c^{\frac {3}{2}} x & \text {for}\: d = 0 \\\frac {2 a^{2} d \sqrt {a d + b d x + c}}{5 b} + \frac {4 a d x \sqrt {a d + b d x + c}}{5} + \frac {4 a c \sqrt {a d + b d x + c}}{5 b} + \frac {2 b d x^{2} \sqrt {a d + b d x + c}}{5} + \frac {4 c x \sqrt {a d + b d x + c}}{5} + \frac {2 c^{2} \sqrt {a d + b d x + c}}{5 b d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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