3.1996 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^3}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=115 \[ -\frac {6 c^2 d^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^4}+\frac {2 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{e^4}-\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )^3}{e^4}+\frac {2 c^3 d^3 (d+e x)^{7/2}}{7 e^4} \]

[Out]

2*c*d*(-a*e^2+c*d^2)^2*(e*x+d)^(3/2)/e^4-6/5*c^2*d^2*(-a*e^2+c*d^2)*(e*x+d)^(5/2)/e^4+2/7*c^3*d^3*(e*x+d)^(7/2
)/e^4-2*(-a*e^2+c*d^2)^3*(e*x+d)^(1/2)/e^4

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Rubi [A]  time = 0.05, antiderivative size = 115, 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} \[ -\frac {6 c^2 d^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^4}+\frac {2 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{e^4}-\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )^3}{e^4}+\frac {2 c^3 d^3 (d+e x)^{7/2}}{7 e^4} \]

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)^(7/2),x]

[Out]

(-2*(c*d^2 - a*e^2)^3*Sqrt[d + e*x])/e^4 + (2*c*d*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2))/e^4 - (6*c^2*d^2*(c*d^2 -
a*e^2)*(d + e*x)^(5/2))/(5*e^4) + (2*c^3*d^3*(d + e*x)^(7/2))/(7*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)^{7/2}} \, dx &=\int \frac {(a e+c d x)^3}{\sqrt {d+e x}} \, dx\\ &=\int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 \sqrt {d+e x}}+\frac {3 c d \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{e^3}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{e^3}+\frac {c^3 d^3 (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac {2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{e^4}+\frac {2 c d \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{e^4}-\frac {6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 e^4}+\frac {2 c^3 d^3 (d+e x)^{7/2}}{7 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 110, normalized size = 0.96 \[ \frac {2 \sqrt {d+e x} \left (35 a^3 e^6+35 a^2 c d e^4 (e x-2 d)+7 a c^2 d^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^3 d^3 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )}{35 e^4} \]

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)^(7/2),x]

[Out]

(2*Sqrt[d + e*x]*(35*a^3*e^6 + 35*a^2*c*d*e^4*(-2*d + e*x) + 7*a*c^2*d^2*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + c
^3*d^3*(-16*d^3 + 8*d^2*e*x - 6*d*e^2*x^2 + 5*e^3*x^3)))/(35*e^4)

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fricas [A]  time = 0.94, size = 130, normalized size = 1.13 \[ \frac {2 \, {\left (5 \, c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 56 \, a c^{2} d^{4} e^{2} - 70 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} - 3 \, {\left (2 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (8 \, c^{3} d^{5} e - 28 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{35 \, e^{4}} \]

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)^(7/2),x, algorithm="fricas")

[Out]

2/35*(5*c^3*d^3*e^3*x^3 - 16*c^3*d^6 + 56*a*c^2*d^4*e^2 - 70*a^2*c*d^2*e^4 + 35*a^3*e^6 - 3*(2*c^3*d^4*e^2 - 7
*a*c^2*d^2*e^4)*x^2 + (8*c^3*d^5*e - 28*a*c^2*d^3*e^3 + 35*a^2*c*d*e^5)*x)*sqrt(e*x + d)/e^4

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giac [A]  time = 0.31, size = 185, normalized size = 1.61 \[ \frac {2}{35} \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{3} e^{24} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{4} e^{24} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{5} e^{24} - 35 \, \sqrt {x e + d} c^{3} d^{6} e^{24} + 21 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{26} - 70 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{26} + 105 \, \sqrt {x e + d} a c^{2} d^{4} e^{26} + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d e^{28} - 105 \, \sqrt {x e + d} a^{2} c d^{2} e^{28} + 35 \, \sqrt {x e + d} a^{3} e^{30}\right )} e^{\left (-28\right )} \]

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)^(7/2),x, algorithm="giac")

[Out]

2/35*(5*(x*e + d)^(7/2)*c^3*d^3*e^24 - 21*(x*e + d)^(5/2)*c^3*d^4*e^24 + 35*(x*e + d)^(3/2)*c^3*d^5*e^24 - 35*
sqrt(x*e + d)*c^3*d^6*e^24 + 21*(x*e + d)^(5/2)*a*c^2*d^2*e^26 - 70*(x*e + d)^(3/2)*a*c^2*d^3*e^26 + 105*sqrt(
x*e + d)*a*c^2*d^4*e^26 + 35*(x*e + d)^(3/2)*a^2*c*d*e^28 - 105*sqrt(x*e + d)*a^2*c*d^2*e^28 + 35*sqrt(x*e + d
)*a^3*e^30)*e^(-28)

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maple [A]  time = 0.06, size = 131, normalized size = 1.14 \[ \frac {2 \sqrt {e x +d}\, \left (5 c^{3} d^{3} e^{3} x^{3}+21 a \,c^{2} d^{2} e^{4} x^{2}-6 c^{3} d^{4} e^{2} x^{2}+35 a^{2} c d \,e^{5} x -28 a \,c^{2} d^{3} e^{3} x +8 c^{3} d^{5} e x +35 a^{3} e^{6}-70 a^{2} c \,d^{2} e^{4}+56 a \,c^{2} d^{4} e^{2}-16 c^{3} d^{6}\right )}{35 e^{4}} \]

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)^(7/2),x)

[Out]

2/35*(e*x+d)^(1/2)*(5*c^3*d^3*e^3*x^3+21*a*c^2*d^2*e^4*x^2-6*c^3*d^4*e^2*x^2+35*a^2*c*d*e^5*x-28*a*c^2*d^3*e^3
*x+8*c^3*d^5*e*x+35*a^3*e^6-70*a^2*c*d^2*e^4+56*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 = 1.19 \[ \frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} d^{3} - 21 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 35 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {e x + d}\right )}}{35 \, e^{4}} \]

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)^(7/2),x, algorithm="maxima")

[Out]

2/35*(5*(e*x + d)^(7/2)*c^3*d^3 - 21*(c^3*d^4 - a*c^2*d^2*e^2)*(e*x + d)^(5/2) + 35*(c^3*d^5 - 2*a*c^2*d^3*e^2
+ a^2*c*d*e^4)*(e*x + d)^(3/2) - 35*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*sqrt(e*x + d))/e^
4

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mupad [B]  time = 0.07, size = 106, normalized size = 0.92 \[ \frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,\sqrt {d+e\,x}}{e^4}-\frac {\left (6\,c^3\,d^4-6\,a\,c^2\,d^2\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^4}+\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,c\,d\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{e^4} \]

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)^(7/2),x)

[Out]

(2*(a*e^2 - c*d^2)^3*(d + e*x)^(1/2))/e^4 - ((6*c^3*d^4 - 6*a*c^2*d^2*e^2)*(d + e*x)^(5/2))/(5*e^4) + (2*c^3*d
^3*(d + e*x)^(7/2))/(7*e^4) + (2*c*d*(a*e^2 - c*d^2)^2*(d + e*x)^(3/2))/e^4

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**(7/2),x)

[Out]

Timed out

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