3.20.77 \(\int \sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\) [1977]

Optimal. Leaf size=43 \[ \frac {2}{5} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{5/2}+\frac {2 c d (d+e x)^{7/2}}{7 e^2} \]

[Out]

2/5*(a-c*d^2/e^2)*(e*x+d)^(5/2)+2/7*c*d*(e*x+d)^(7/2)/e^2

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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \begin {gather*} \frac {2}{5} (d+e x)^{5/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{7/2}}{7 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(2*(a - (c*d^2)/e^2)*(d + e*x)^(5/2))/5 + (2*c*d*(d + e*x)^(7/2))/(7*e^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])

Rule 640

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

Rubi steps

\begin {align*} \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^{3/2} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) (d+e x)^{3/2}}{e}+\frac {c d (d+e x)^{5/2}}{e}\right ) \, dx\\ &=\frac {2}{5} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{5/2}+\frac {2 c d (d+e x)^{7/2}}{7 e^2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 34, normalized size = 0.79 \begin {gather*} \frac {2 (d+e x)^{5/2} \left (7 a e^2+c d (-2 d+5 e x)\right )}{35 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(2*(d + e*x)^(5/2)*(7*a*e^2 + c*d*(-2*d + 5*e*x)))/(35*e^2)

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Maple [A]
time = 0.46, size = 39, normalized size = 0.91

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (5 c d e x +7 e^{2} a -2 c \,d^{2}\right )}{35 e^{2}}\)	\(32\)
derivativedivides	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{2}}\)	\(39\)
default	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{2}}\)	\(39\)
trager	\(\frac {2 \left (5 d \,e^{3} c \,x^{3}+7 e^{4} a \,x^{2}+8 d^{2} e^{2} c \,x^{2}+14 a d \,e^{3} x +c \,d^{3} e x +7 a \,d^{2} e^{2}-2 c \,d^{4}\right ) \sqrt {e x +d}}{35 e^{2}}\)	\(75\)
risch	\(\frac {2 \left (5 d \,e^{3} c \,x^{3}+7 e^{4} a \,x^{2}+8 d^{2} e^{2} c \,x^{2}+14 a d \,e^{3} x +c \,d^{3} e x +7 a \,d^{2} e^{2}-2 c \,d^{4}\right ) \sqrt {e x +d}}{35 e^{2}}\)	\(75\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e^2*(1/7*c*d*(e*x+d)^(7/2)+1/5*(a*e^2-c*d^2)*(e*x+d)^(5/2))

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Maxima [A]
time = 0.29, size = 38, normalized size = 0.88 \begin {gather*} \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c d - 7 \, {\left (c d^{2} - a e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*(x*e + d)^(7/2)*c*d - 7*(c*d^2 - a*e^2)*(x*e + d)^(5/2))*e^(-2)

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Fricas [A]
time = 2.63, size = 70, normalized size = 1.63 \begin {gather*} \frac {2}{35} \, {\left (c d^{3} x e - 2 \, c d^{4} + 7 \, a x^{2} e^{4} + {\left (5 \, c d x^{3} + 14 \, a d x\right )} e^{3} + {\left (8 \, c d^{2} x^{2} + 7 \, a d^{2}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(c*d^3*x*e - 2*c*d^4 + 7*a*x^2*e^4 + (5*c*d*x^3 + 14*a*d*x)*e^3 + (8*c*d^2*x^2 + 7*a*d^2)*e^2)*sqrt(x*e +
d)*e^(-2)

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Sympy [A]
time = 1.30, size = 41, normalized size = 0.95 \begin {gather*} \frac {2 \left (\frac {c d \left (d + e x\right )^{\frac {7}{2}}}{7 e} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a e^{2} - c d^{2}\right )}{5 e}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(c*d*(d + e*x)**(7/2)/(7*e) + (d + e*x)**(5/2)*(a*e**2 - c*d**2)/(5*e))/e

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (36) = 72\).
time = 1.81, size = 212, normalized size = 4.93 \begin {gather*} \frac {2}{105} \, {\left (35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} c d^{3} e^{\left (-1\right )} + 14 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c d^{2} e^{\left (-1\right )} + 105 \, \sqrt {x e + d} a d^{2} e + 70 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a d e + 3 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c d e^{\left (-1\right )} + 7 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a e\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*c*d^3*e^(-1) + 14*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*c*d^2*e^(-1) + 105*sqrt(x*e + d)*a*d^2*e + 70*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*d*
e + 3*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*c*d*e^(-1) +
7*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*e)*e^(-1)

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Mupad [B]
time = 0.04, size = 34, normalized size = 0.79 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (7\,a\,e^2-7\,c\,d^2+5\,c\,d\,\left (d+e\,x\right )\right )}{35\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(1/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2),x)

[Out]

(2*(d + e*x)^(5/2)*(7*a*e^2 - 7*c*d^2 + 5*c*d*(d + e*x)))/(35*e^2)

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