3.20.86 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^2}{(d+e x)^{3/2}} \, dx\) [1986]

Optimal. Leaf size=83 \[ \frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 e^3}+\frac {2 c^2 d^2 (d+e x)^{7/2}}{7 e^3} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 83, 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)^{5/2} \left (c d^2-a e^2\right )}{5 e^3}+\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 e^3}+\frac {2 c^2 d^2 (d+e x)^{7/2}}{7 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)^(3/2),x]

[Out]

(2*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2))/(3*e^3) - (4*c*d*(c*d^2 - a*e^2)*(d + e*x)^(5/2))/(5*e^3) + (2*c^2*d^2*(
d + e*x)^(7/2))/(7*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)^{3/2}} \, dx &=\int (a e+c d x)^2 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^2 \sqrt {d+e x}}{e^2}-\frac {2 c d \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{e^2}+\frac {c^2 d^2 (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 e^3}+\frac {2 c^2 d^2 (d+e x)^{7/2}}{7 e^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 67, normalized size = 0.81 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (35 a^2 e^4+14 a c d e^2 (-2 d+3 e x)+c^2 d^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 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)^(3/2),x]

[Out]

(2*(d + e*x)^(3/2)*(35*a^2*e^4 + 14*a*c*d*e^2*(-2*d + 3*e*x) + c^2*d^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*
e^3)

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Maple [A]
time = 0.69, size = 68, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (e^{2} a -c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\)	\(68\)
default	\(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (e^{2} a -c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\)	\(68\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (15 e^{2} x^{2} c^{2} d^{2}+42 a c d \,e^{3} x -12 c^{2} d^{3} e x +35 a^{2} e^{4}-28 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{105 e^{3}}\)	\(73\)
trager	\(\frac {2 \left (15 c^{2} d^{2} e^{3} x^{3}+42 a c d \,e^{4} x^{2}+3 c^{2} d^{3} e^{2} x^{2}+35 a^{2} e^{5} x +14 a c \,e^{3} d^{2} x -4 c^{2} d^{4} e x +35 a^{2} d \,e^{4}-28 a c \,d^{3} e^{2}+8 c^{2} d^{5}\right ) \sqrt {e x +d}}{105 e^{3}}\)	\(110\)
risch	\(\frac {2 \left (15 c^{2} d^{2} e^{3} x^{3}+42 a c d \,e^{4} x^{2}+3 c^{2} d^{3} e^{2} x^{2}+35 a^{2} e^{5} x +14 a c \,e^{3} d^{2} x -4 c^{2} d^{4} e x +35 a^{2} d \,e^{4}-28 a c \,d^{3} e^{2}+8 c^{2} d^{5}\right ) \sqrt {e x +d}}{105 e^{3}}\)	\(110\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/e^3*(1/7*c^2*d^2*(e*x+d)^(7/2)+2/5*(a*e^2-c*d^2)*c*d*(e*x+d)^(5/2)+1/3*(a*e^2-c*d^2)^2*(e*x+d)^(3/2))

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Maxima [A]
time = 0.29, size = 79, normalized size = 0.95 \begin {gather*} \frac {2}{105} \, {\left (15 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d^{2} - 42 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (x e + d\right )}^{\frac {5}{2}} + 35 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}\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)^(3/2),x, algorithm="maxima")

[Out]

2/105*(15*(x*e + d)^(7/2)*c^2*d^2 - 42*(c^2*d^3 - a*c*d*e^2)*(x*e + d)^(5/2) + 35*(c^2*d^4 - 2*a*c*d^2*e^2 + a
^2*e^4)*(x*e + d)^(3/2))*e^(-3)

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Fricas [A]
time = 3.97, size = 106, normalized size = 1.28 \begin {gather*} -\frac {2}{105} \, {\left (4 \, c^{2} d^{4} x e - 8 \, c^{2} d^{5} - 35 \, a^{2} x e^{5} - 7 \, {\left (6 \, a c d x^{2} + 5 \, a^{2} d\right )} e^{4} - {\left (15 \, c^{2} d^{2} x^{3} + 14 \, a c d^{2} x\right )} e^{3} - {\left (3 \, c^{2} d^{3} x^{2} - 28 \, a c d^{3}\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)^(3/2),x, algorithm="fricas")

[Out]

-2/105*(4*c^2*d^4*x*e - 8*c^2*d^5 - 35*a^2*x*e^5 - 7*(6*a*c*d*x^2 + 5*a^2*d)*e^4 - (15*c^2*d^2*x^3 + 14*a*c*d^
2*x)*e^3 - (3*c^2*d^3*x^2 - 28*a*c*d^3)*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. 411 vs. \(2 (76) = 152\).
time = 19.75, size = 411, normalized size = 4.95 \begin {gather*} \begin {cases} \frac {- \frac {2 a^{2} d^{2} e^{2}}{\sqrt {d + e x}} - 4 a^{2} d e^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 a^{2} e^{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 ) - 4 a c d^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 8 a 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 ) - 4 a 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 ) - \frac {2 c^{2} d^{4} \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 {4 c^{2} d^{3} \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}} - \frac {2 c^{2} d^{2} \left (\frac {d^{4}}{\sqrt {d + e x}} + 4 d^{3} \sqrt {d + e x} - 2 d^{2} \left (d + e x\right )^{\frac {3}{2}} + \frac {4 d \left (d + e x\right )^{\frac {5}{2}}}{5} - \frac {\left (d + e x\right )^{\frac {7}{2}}}{7}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {5}{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)**(3/2),x)

[Out]

Piecewise(((-2*a**2*d**2*e**2/sqrt(d + e*x) - 4*a**2*d*e**2*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 2*a**2*e**2*(
d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 4*a*c*d**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) -
8*a*c*d**2*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 4*a*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) - 2*c**2*d**4*(d**2/sqrt(d + e*x) + 2*d*sqrt(d +
e*x) - (d + e*x)**(3/2)/3)/e**2 - 4*c**2*d**3*(-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 - 2*c**2*d**2*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2*d**2*(d + e*x)**(3/2)
+ 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**2)/e, Ne(e, 0)), (c**2*d**(5/2)*x**3/3, True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (69) = 138\).
time = 1.24, size = 218, normalized size = 2.63 \begin {gather*} \frac {2}{105} \, {\left (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 )} c^{2} d^{3} e^{\left (-2\right )} + 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^{2} d^{2} e^{\left (-2\right )} + 70 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a c d^{2} + 105 \, \sqrt {x e + d} a^{2} d e^{2} + 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 )} a c d + 35 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} 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)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.63, size = 80, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (35\,a^2\,e^4+35\,c^2\,d^4+15\,c^2\,d^2\,{\left (d+e\,x\right )}^2-42\,c^2\,d^3\,\left (d+e\,x\right )-70\,a\,c\,d^2\,e^2+42\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{105\,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)^(3/2),x)

[Out]

(2*(d + e*x)^(3/2)*(35*a^2*e^4 + 35*c^2*d^4 + 15*c^2*d^2*(d + e*x)^2 - 42*c^2*d^3*(d + e*x) - 70*a*c*d^2*e^2 +
42*a*c*d*e^2*(d + e*x)))/(105*e^3)

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