3.1995 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=119 \[ -\frac {6 c^2 d^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^4}+\frac {6 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^4}-\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}{3 e^4}+\frac {2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \]

[Out]

-2/3*(-a*e^2+c*d^2)^3*(e*x+d)^(3/2)/e^4+6/5*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^(5/2)/e^4-6/7*c^2*d^2*(-a*e^2+c*d^2)*
(e*x+d)^(7/2)/e^4+2/9*c^3*d^3*(e*x+d)^(9/2)/e^4

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Rubi [A]  time = 0.05, antiderivative size = 119, 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 = {626, 43} \[ -\frac {6 c^2 d^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^4}+\frac {6 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^4}-\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}{3 e^4}+\frac {2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^3*(d + e*x)^(3/2))/(3*e^4) + (6*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(5/2))/(5*e^4) - (6*c^2*d^
2*(c*d^2 - a*e^2)*(d + e*x)^(7/2))/(7*e^4) + (2*c^3*d^3*(d + e*x)^(9/2))/(9*e^4)

Rule 43

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 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^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 )^3}{(d+e x)^{5/2}} \, dx &=\int (a e+c d x)^3 \sqrt {d+e x} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3 \sqrt {d+e x}}{e^3}+\frac {3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{e^3}+\frac {c^3 d^3 (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}{3 e^4}+\frac {6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^4}+\frac {2 c^3 d^3 (d+e x)^{9/2}}{9 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 98, normalized size = 0.82 \[ \frac {2 (d+e x)^{3/2} \left (-135 c^2 d^2 (d+e x)^2 \left (c d^2-a e^2\right )+189 c d (d+e x) \left (c d^2-a e^2\right )^2-105 \left (c d^2-a e^2\right )^3+35 c^3 d^3 (d+e x)^3\right )}{315 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3/(d + e*x)^(5/2),x]

[Out]

(2*(d + e*x)^(3/2)*(-105*(c*d^2 - a*e^2)^3 + 189*c*d*(c*d^2 - a*e^2)^2*(d + e*x) - 135*c^2*d^2*(c*d^2 - a*e^2)
*(d + e*x)^2 + 35*c^3*d^3*(d + e*x)^3))/(315*e^4)

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fricas [A]  time = 0.76, size = 179, normalized size = 1.50 \[ \frac {2 \, {\left (35 \, c^{3} d^{3} e^{4} x^{4} - 16 \, c^{3} d^{7} + 72 \, a c^{2} d^{5} e^{2} - 126 \, a^{2} c d^{3} e^{4} + 105 \, a^{3} d e^{6} + 5 \, {\left (c^{3} d^{4} e^{3} + 27 \, a c^{2} d^{2} e^{5}\right )} x^{3} - 3 \, {\left (2 \, c^{3} d^{5} e^{2} - 9 \, a c^{2} d^{3} e^{4} - 63 \, a^{2} c d e^{6}\right )} x^{2} + {\left (8 \, c^{3} d^{6} e - 36 \, a c^{2} d^{4} e^{3} + 63 \, a^{2} c d^{2} e^{5} + 105 \, a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{4}} \]

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)^3/(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/315*(35*c^3*d^3*e^4*x^4 - 16*c^3*d^7 + 72*a*c^2*d^5*e^2 - 126*a^2*c*d^3*e^4 + 105*a^3*d*e^6 + 5*(c^3*d^4*e^3
+ 27*a*c^2*d^2*e^5)*x^3 - 3*(2*c^3*d^5*e^2 - 9*a*c^2*d^3*e^4 - 63*a^2*c*d*e^6)*x^2 + (8*c^3*d^6*e - 36*a*c^2*
d^4*e^3 + 63*a^2*c*d^2*e^5 + 105*a^3*e^7)*x)*sqrt(e*x + d)/e^4

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giac [A]  time = 0.35, size = 185, normalized size = 1.55 \[ \frac {2}{315} \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} c^{3} d^{3} e^{32} - 135 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{4} e^{32} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{5} e^{32} - 105 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{6} e^{32} + 135 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} d^{2} e^{34} - 378 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{3} e^{34} + 315 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{4} e^{34} + 189 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} c d e^{36} - 315 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d^{2} e^{36} + 105 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{3} e^{38}\right )} e^{\left (-36\right )} \]

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)^3/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/315*(35*(x*e + d)^(9/2)*c^3*d^3*e^32 - 135*(x*e + d)^(7/2)*c^3*d^4*e^32 + 189*(x*e + d)^(5/2)*c^3*d^5*e^32 -
105*(x*e + d)^(3/2)*c^3*d^6*e^32 + 135*(x*e + d)^(7/2)*a*c^2*d^2*e^34 - 378*(x*e + d)^(5/2)*a*c^2*d^3*e^34 +
315*(x*e + d)^(3/2)*a*c^2*d^4*e^34 + 189*(x*e + d)^(5/2)*a^2*c*d*e^36 - 315*(x*e + d)^(3/2)*a^2*c*d^2*e^36 + 1
05*(x*e + d)^(3/2)*a^3*e^38)*e^(-36)

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maple [A]  time = 0.05, size = 131, normalized size = 1.10 \[ \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 c^{3} d^{3} e^{3} x^{3}+135 a \,c^{2} d^{2} e^{4} x^{2}-30 c^{3} d^{4} e^{2} x^{2}+189 a^{2} c d \,e^{5} x -108 a \,c^{2} d^{3} e^{3} x +24 c^{3} d^{5} e x +105 a^{3} e^{6}-126 a^{2} c \,d^{2} e^{4}+72 a \,c^{2} d^{4} e^{2}-16 c^{3} d^{6}\right )}{315 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^3/(e*x+d)^(5/2),x)

[Out]

2/315*(e*x+d)^(3/2)*(35*c^3*d^3*e^3*x^3+135*a*c^2*d^2*e^4*x^2-30*c^3*d^4*e^2*x^2+189*a^2*c*d*e^5*x-108*a*c^2*d
^3*e^3*x+24*c^3*d^5*e*x+105*a^3*e^6-126*a^2*c*d^2*e^4+72*a*c^2*d^4*e^2-16*c^3*d^6)/e^4

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maxima [A]  time = 1.11, size = 137, normalized size = 1.15 \[ \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} d^{3} - 135 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{315 \, e^{4}} \]

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)^3/(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/315*(35*(e*x + d)^(9/2)*c^3*d^3 - 135*(c^3*d^4 - a*c^2*d^2*e^2)*(e*x + d)^(7/2) + 189*(c^3*d^5 - 2*a*c^2*d^3
*e^2 + a^2*c*d*e^4)*(e*x + d)^(5/2) - 105*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*(e*x + d)^(3
/2))/e^4

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mupad [B]  time = 0.07, size = 106, normalized size = 0.89 \[ \frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {6\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4} \]

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)^3/(d + e*x)^(5/2),x)

[Out]

(2*(a*e^2 - c*d^2)^3*(d + e*x)^(3/2))/(3*e^4) - ((6*c^3*d^4 - 6*a*c^2*d^2*e^2)*(d + e*x)^(7/2))/(7*e^4) + (2*c
^3*d^3*(d + e*x)^(9/2))/(9*e^4) + (6*c*d*(a*e^2 - c*d^2)^2*(d + e*x)^(5/2))/(5*e^4)

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sympy [A]  time = 98.84, size = 644, normalized size = 5.41 \[ \begin {cases} \frac {- \frac {2 a^{3} d^{2} e^{3}}{\sqrt {d + e x}} - 4 a^{3} d e^{3} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 2 a^{3} e^{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 ) - 6 a^{2} c d^{3} e \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - 12 a^{2} c d^{2} 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 ) - 6 a^{2} c d e \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 {6 a 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} - \frac {12 a 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} - \frac {6 a 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} - \frac {2 c^{3} d^{5} \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^{3}} - \frac {4 c^{3} d^{4} \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^{3}} - \frac {2 c^{3} d^{3} \left (- \frac {d^{5}}{\sqrt {d + e x}} - 5 d^{4} \sqrt {d + e x} + \frac {10 d^{3} \left (d + e x\right )^{\frac {3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac {5}{2}} + \frac {5 d \left (d + e x\right )^{\frac {7}{2}}}{7} - \frac {\left (d + e x\right )^{\frac {9}{2}}}{9}\right )}{e^{3}}}{e} & \text {for}\: e \neq 0 \\\frac {c^{3} d^{\frac {7}{2}} x^{4}}{4} & \text {otherwise} \end {cases} \]

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)**3/(e*x+d)**(5/2),x)

[Out]

Piecewise(((-2*a**3*d**2*e**3/sqrt(d + e*x) - 4*a**3*d*e**3*(-d/sqrt(d + e*x) - sqrt(d + e*x)) - 2*a**3*e**3*(
d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 6*a**2*c*d**3*e*(-d/sqrt(d + e*x) - sqrt(d + e*
x)) - 12*a**2*c*d**2*e*(d**2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3) - 6*a**2*c*d*e*(-d**3/sqr
t(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5) - 6*a*c**2*d**4*(d**2/sqrt(d + e*
x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e - 12*a*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 - 6*a*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*c**3*d**5*(-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**3 - 4*c**3*d**4*(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**3 - 2*c**3*d**3*(-d**5/s
qrt(d + e*x) - 5*d**4*sqrt(d + e*x) + 10*d**3*(d + e*x)**(3/2)/3 - 2*d**2*(d + e*x)**(5/2) + 5*d*(d + e*x)**(7
/2)/7 - (d + e*x)**(9/2)/9)/e**3)/e, Ne(e, 0)), (c**3*d**(7/2)*x**4/4, True))

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