3.20.78 \(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{\sqrt {d+e x}} \, dx\) [1978]

Optimal. Leaf size=43 \[ \frac {2}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{3/2}+\frac {2 c d (d+e x)^{5/2}}{5 e^2} \]

[Out]

2/3*(a-c*d^2/e^2)*(e*x+d)^(3/2)+2/5*c*d*(e*x+d)^(5/2)/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}{3} (d+e x)^{3/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{5/2}}{5 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a - (c*d^2)/e^2)*(d + e*x)^(3/2))/3 + (2*c*d*(d + e*x)^(5/2))/(5*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 \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx &=\int (a e+c d x) \sqrt {d+e x} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right ) \sqrt {d+e x}}{e}+\frac {c d (d+e x)^{3/2}}{e}\right ) \, dx\\ &=\frac {2}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{3/2}+\frac {2 c d (d+e x)^{5/2}}{5 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)^{3/2} \left (5 a e^2+c d (-2 d+3 e x)\right )}{15 e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.47, size = 39, normalized size = 0.91

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3 c d e x +5 e^{2} a -2 c \,d^{2}\right )}{15 e^{2}}\)	\(32\)
derivativedivides	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{2}}\)	\(39\)
default	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{2}}\)	\(39\)
trager	\(\frac {2 \left (3 c d \,e^{2} x^{2}+5 a \,e^{3} x +c \,d^{2} e x +5 a d \,e^{2}-2 c \,d^{3}\right ) \sqrt {e x +d}}{15 e^{2}}\)	\(51\)
risch	\(\frac {2 \left (3 c d \,e^{2} x^{2}+5 a \,e^{3} x +c \,d^{2} e x +5 a d \,e^{2}-2 c \,d^{3}\right ) \sqrt {e x +d}}{15 e^{2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (36) = 72\).
time = 0.28, size = 93, normalized size = 2.16 \begin {gather*} \frac {2}{15} \, {\left ({\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 e^{\left (-1\right )} + 15 \, \sqrt {x e + d} a d e + 5 \, {\left (c d^{2} + a e^{2}\right )} {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} e^{\left (-1\right )}\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*((3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*c*d*e^(-1) + 15*sqrt(x*e + d)*a*d*e +
5*(c*d^2 + a*e^2)*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*e^(-1))*e^(-1)

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Fricas [A]
time = 3.42, size = 48, normalized size = 1.12 \begin {gather*} \frac {2}{15} \, {\left (c d^{2} x e - 2 \, c d^{3} + 5 \, a x e^{3} + {\left (3 \, c d x^{2} + 5 \, a d\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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (42) = 84\).
time = 11.41, size = 221, normalized size = 5.14 \begin {gather*} \begin {cases} \frac {- \frac {2 a d^{2} e}{\sqrt {d + e x}} - 4 a d e \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 a e \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right ) - \frac {2 c d^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {4 c d^{2} \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e} - \frac {2 c d \left (- \frac {d^{3}}{\sqrt {d + e x}} - 3 d^{2} \sqrt {d + e x} + d \left (d + e x\right )^{\frac {3}{2}} - \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e}}{e} & \text {for}\: e \neq 0 \\\frac {c d^{\frac {3}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-2*a*d**2*e/sqrt(d + e*x) - 4*a*d*e*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 2*a*e*(d**2/sqrt(d + e*x)
+ 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 2*c*d**3*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e - 4*c*d**2*(d**2/sq
rt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 2*c*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) +
d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e)/e, Ne(e, 0)), (c*d**(3/2)*x**2/2, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (36) = 72\).
time = 0.97, size = 112, normalized size = 2.60 \begin {gather*} \frac {2}{15} \, {\left (5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} c d^{2} e^{\left (-1\right )} + {\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 e^{\left (-1\right )} + 15 \, \sqrt {x e + d} a d e + 5 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a e\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(5*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*c*d^2*e^(-1) + (3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sq
rt(x*e + d)*d^2)*c*d*e^(-1) + 15*sqrt(x*e + d)*a*d*e + 5*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a*e)*e^(-1)

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Mupad [B]
time = 0.05, size = 34, normalized size = 0.79 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (5\,a\,e^2-5\,c\,d^2+3\,c\,d\,\left (d+e\,x\right )\right )}{15\,e^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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