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*(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)

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Rubi [A]  time = 0.051909, antiderivative size = 119, normalized size of antiderivative = 1., 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 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]

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])

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*}

Mathematica [A]  time = 0.0666193, 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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Maple [A]  time = 0.048, size = 131, normalized size = 1.1 \begin{align*}{\frac{70\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+270\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-60\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+378\,{a}^{2}cd{e}^{5}x-216\,a{c}^{2}{d}^{3}{e}^{3}x+48\,{c}^{3}{d}^{5}ex+210\,{a}^{3}{e}^{6}-252\,{a}^{2}c{d}^{2}{e}^{4}+144\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{315\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

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)^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 = 1.0738, size = 185, normalized size = 1.55 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.8748, size = 383, normalized size = 3.22 \begin{align*} \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}} \end{align*}

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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Sympy [A]  time = 102.341, size = 644, normalized size = 5.41 \begin{align*} \text{result too large to display} \end{align*}

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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Giac [A]  time = 1.20444, size = 250, normalized size = 2.1 \begin{align*} \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 )} \end{align*}

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)