3.14.79 \(\int (a+b x) \sqrt {c+d x} \, dx\) [1379]

Optimal. Leaf size=42 \[ -\frac {2 (b c-a d) (c+d x)^{3/2}}{3 d^2}+\frac {2 b (c+d x)^{5/2}}{5 d^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \begin {gather*} \frac {2 b (c+d x)^{5/2}}{5 d^2}-\frac {2 (c+d x)^{3/2} (b c-a d)}{3 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*Sqrt[c + d*x],x]

[Out]

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

Rule 45

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

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Mathematica [A]
time = 0.02, size = 30, normalized size = 0.71 \begin {gather*} \frac {2 (c+d x)^{3/2} (-2 b c+5 a d+3 b d x)}{15 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*Sqrt[c + d*x],x]

[Out]

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

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Mathics [A]
time = 2.74, size = 29, normalized size = 0.69 \begin {gather*} \frac {2 \left (5 a d+3 b \left (c+d x\right )-5 b c\right ) \left (c+d x\right )^{\frac {3}{2}}}{15 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^1*(c + d*x)^(1/2),x]')

[Out]

2 (5 a d + 3 b (c + d x) - 5 b c) (c + d x) ^ (3 / 2) / (15 d ^ 2)

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Maple [A]
time = 0.12, size = 34, normalized size = 0.81

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (3 b d x +5 a d -2 b c \right )}{15 d^{2}}\)	\(27\)
derivativedivides	\(\frac {\frac {2 b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{2}}\)	\(34\)
default	\(\frac {\frac {2 b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{2}}\)	\(34\)
trager	\(\frac {2 \left (3 b \,d^{2} x^{2}+5 a \,d^{2} x +b c d x +5 a c d -2 b \,c^{2}\right ) \sqrt {d x +c}}{15 d^{2}}\)	\(46\)
risch	\(\frac {2 \left (3 b \,d^{2} x^{2}+5 a \,d^{2} x +b c d x +5 a c d -2 b \,c^{2}\right ) \sqrt {d x +c}}{15 d^{2}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d^2*(1/5*b*(d*x+c)^(5/2)+1/3*(a*d-b*c)*(d*x+c)^(3/2))

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Maxima [A]
time = 0.27, size = 33, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} b - 5 \, {\left (b c - a d\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{15 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.31, size = 46, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (3 \, b d^{2} x^{2} - 2 \, b c^{2} + 5 \, a c d + {\left (b c d + 5 \, a d^{2}\right )} x\right )} \sqrt {d x + c}}{15 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.92, size = 36, normalized size = 0.86 \begin {gather*} \frac {2 \left (\frac {b \left (c + d x\right )^{\frac {5}{2}}}{5 d} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a d - b c\right )}{3 d}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (34) = 68\).
time = 0.00, size = 147, normalized size = 3.50 \begin {gather*} \frac {\frac {2 b d \left (\frac {1}{5} \sqrt {c+d x} \left (c+d x\right )^{2}-\frac {2}{3} \sqrt {c+d x} \left (c+d x\right ) c+\sqrt {c+d x} c^{2}\right )}{d^{2}}+2 a \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )+\frac {2 b c \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )}{d}+2 a c \sqrt {c+d x}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 29, normalized size = 0.69 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{3/2}\,\left (5\,a\,d-5\,b\,c+3\,b\,\left (c+d\,x\right )\right )}{15\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(c + d*x)^(1/2),x)

[Out]

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

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