3.20.87 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{5/2}} \, dx\) [1987]

Optimal. Leaf size=81 \[ \frac {2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 e^3}+\frac {2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \]

[Out]

-4/3*c*d*(-a*e^2+c*d^2)*(e*x+d)^(3/2)/e^3+2/5*c^2*d^2*(e*x+d)^(5/2)/e^3+2*(-a*e^2+c*d^2)^2*(e*x+d)^(1/2)/e^3

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Rubi [A]
time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {640, 45} \begin {gather*} -\frac {4 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{e^3}+\frac {2 c^2 d^2 (d+e x)^{5/2}}{5 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(5/2),x]

[Out]

(2*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/e^3 - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)^(3/2))/(3*e^3) + (2*c^2*d^2*(d + e*
x)^(5/2))/(5*e^3)

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 {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \frac {(a e+c d x)^2}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2}{e^2 \sqrt {d+e x}}-\frac {2 c d \left (c d^2-a e^2\right ) \sqrt {d+e x}}{e^2}+\frac {c^2 d^2 (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 e^3}+\frac {2 c^2 d^2 (d+e x)^{5/2}}{5 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 66, normalized size = 0.81 \begin {gather*} \frac {2 \sqrt {d+e x} \left (15 a^2 e^4+10 a c d e^2 (-2 d+e x)+c^2 d^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(15*a^2*e^4 + 10*a*c*d*e^2*(-2*d + e*x) + c^2*d^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

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Maple [A]
time = 0.70, size = 67, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (e^{2} a -c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{3}}\)	\(67\)
default	\(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (e^{2} a -c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (e^{2} a -c \,d^{2}\right )^{2} \sqrt {e x +d}}{e^{3}}\)	\(67\)
gosper	\(\frac {2 \sqrt {e x +d}\, \left (3 e^{2} x^{2} c^{2} d^{2}+10 a c d \,e^{3} x -4 c^{2} d^{3} e x +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 e^{3}}\)	\(73\)
trager	\(\frac {2 \sqrt {e x +d}\, \left (3 e^{2} x^{2} c^{2} d^{2}+10 a c d \,e^{3} x -4 c^{2} d^{3} e x +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 e^{3}}\)	\(73\)
risch	\(\frac {2 \sqrt {e x +d}\, \left (3 e^{2} x^{2} c^{2} d^{2}+10 a c d \,e^{3} x -4 c^{2} d^{3} e x +15 a^{2} e^{4}-20 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{15 e^{3}}\)	\(73\)

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)^2/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 79, normalized size = 0.98 \begin {gather*} \frac {2}{15} \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{2} - 10 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (x e + d\right )}^{\frac {3}{2}} + 15 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {x e + d}\right )} e^{\left (-3\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)^2/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/15*(3*(x*e + d)^(5/2)*c^2*d^2 - 10*(c^2*d^3 - a*c*d*e^2)*(x*e + d)^(3/2) + 15*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2
*e^4)*sqrt(x*e + d))*e^(-3)

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Fricas [A]
time = 3.07, size = 70, normalized size = 0.86 \begin {gather*} -\frac {2}{15} \, {\left (4 \, c^{2} d^{3} x e - 8 \, c^{2} d^{4} - 10 \, a c d x e^{3} - 15 \, a^{2} e^{4} - {\left (3 \, c^{2} d^{2} x^{2} - 20 \, a c d^{2}\right )} e^{2}\right )} \sqrt {x e + d} e^{\left (-3\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)^2/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

-2/15*(4*c^2*d^3*x*e - 8*c^2*d^4 - 10*a*c*d*x*e^3 - 15*a^2*e^4 - (3*c^2*d^2*x^2 - 20*a*c*d^2)*e^2)*sqrt(x*e +
d)*e^(-3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (75) = 150\).
time = 19.06, size = 236, normalized size = 2.91 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{2} d e^{2}}{\sqrt {d + e x}} - 2 a^{2} e^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 4 a c d^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 4 a c d \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^{2} d^{3} \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^{2}} - \frac {2 c^{2} d^{2} \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^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {3}{2}} x^{3}}{3} & \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)**2/(e*x+d)**(5/2),x)

[Out]

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

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Giac [A]
time = 1.12, size = 89, normalized size = 1.10 \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^{2} d^{2} e^{\left (-2\right )} + 10 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a c d + 15 \, \sqrt {x e + d} a^{2} e^{2}\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)^2/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.06, size = 80, normalized size = 0.99 \begin {gather*} \frac {2\,\sqrt {d+e\,x}\,\left (15\,a^2\,e^4+15\,c^2\,d^4+3\,c^2\,d^2\,{\left (d+e\,x\right )}^2-10\,c^2\,d^3\,\left (d+e\,x\right )-30\,a\,c\,d^2\,e^2+10\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{15\,e^3} \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)^2/(d + e*x)^(5/2),x)

[Out]

(2*(d + e*x)^(1/2)*(15*a^2*e^4 + 15*c^2*d^4 + 3*c^2*d^2*(d + e*x)^2 - 10*c^2*d^3*(d + e*x) - 30*a*c*d^2*e^2 +
10*a*c*d*e^2*(d + e*x)))/(15*e^3)

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