3.20.28 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{(d+e x)^6} \, dx\) [1928]

Optimal. Leaf size=111 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^5} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664} \begin {gather*} \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 (d+e x)^5 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 (d+e x)^6 \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

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

Rule 664

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^m*((a +
b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b*e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rule 672

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d -
b*e))), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a
*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 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 )^{3/2}}{(d+e x)^6} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {(2 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^5} \, dx}{7 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^6}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^5}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 61, normalized size = 0.55 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{5/2} \left (-5 a e^2+c d (7 d+2 e x)\right )}{35 \left (c d^2-a e^2\right )^2 (d+e x)^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(d + e*x)^6,x]

[Out]

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

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Maple [A]
time = 0.79, size = 131, normalized size = 1.18

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-2 c d e x +5 e^{2} a -7 c \,d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}{35 \left (e x +d \right )^{5} \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}\)	\(90\)
default	\(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{6}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{35 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{5}}}{e^{6}}\)	\(131\)
trager	\(-\frac {2 \left (-2 c^{3} d^{3} e \,x^{3}+d^{2} e^{2} c^{2} a \,x^{2}-7 c^{3} d^{4} x^{2}+8 a^{2} c d \,e^{3} x -14 d^{3} e \,c^{2} a x +5 e^{4} a^{3}-7 d^{2} e^{2} a^{2} c \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{35 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{4}}\)	\(143\)

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)^(3/2)/(e*x+d)^6,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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)^(3/2)/(e*x+d)^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (105) = 210\).
time = 7.71, size = 275, normalized size = 2.48 \begin {gather*} \frac {2 \, {\left (7 \, c^{3} d^{4} x^{2} - 8 \, a^{2} c d x e^{3} - 5 \, a^{3} e^{4} - {\left (a c^{2} d^{2} x^{2} - 7 \, a^{2} c d^{2}\right )} e^{2} + 2 \, {\left (c^{3} d^{3} x^{3} + 7 \, a c^{2} d^{3} x\right )} e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{35 \, {\left (4 \, c^{2} d^{7} x e + c^{2} d^{8} + a^{2} x^{4} e^{8} + 4 \, a^{2} d x^{3} e^{7} - 2 \, {\left (a c d^{2} x^{4} - 3 \, a^{2} d^{2} x^{2}\right )} e^{6} - 4 \, {\left (2 \, a c d^{3} x^{3} - a^{2} d^{3} x\right )} e^{5} + {\left (c^{2} d^{4} x^{4} - 12 \, a c d^{4} x^{2} + a^{2} d^{4}\right )} e^{4} + 4 \, {\left (c^{2} d^{5} x^{3} - 2 \, a c d^{5} x\right )} e^{3} + 2 \, {\left (3 \, c^{2} d^{6} x^{2} - a c d^{6}\right )} e^{2}\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)^(3/2)/(e*x+d)^6,x, algorithm="fricas")

[Out]

2/35*(7*c^3*d^4*x^2 - 8*a^2*c*d*x*e^3 - 5*a^3*e^4 - (a*c^2*d^2*x^2 - 7*a^2*c*d^2)*e^2 + 2*(c^3*d^3*x^3 + 7*a*c
^2*d^3*x)*e)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)/(4*c^2*d^7*x*e + c^2*d^8 + a^2*x^4*e^8 + 4*a^2*d*x^3*
e^7 - 2*(a*c*d^2*x^4 - 3*a^2*d^2*x^2)*e^6 - 4*(2*a*c*d^3*x^3 - a^2*d^3*x)*e^5 + (c^2*d^4*x^4 - 12*a*c*d^4*x^2
+ a^2*d^4)*e^4 + 4*(c^2*d^5*x^3 - 2*a*c*d^5*x)*e^3 + 2*(3*c^2*d^6*x^2 - a*c*d^6)*e^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**(3/2)/(e*x+d)**6,x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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)^(3/2)/(e*x+d)^6,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{%%%{1,[0,0,4]%%%},[8]%%%}+%%%{%%{[%%%{-8,[0,1,3]%%%},0]:
[1,0,%%%{-1

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Mupad [B]
time = 2.77, size = 1477, normalized size = 13.31 \begin {gather*} \frac {\left (\frac {d\,\left (\frac {4\,c^3\,d^4}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {2\,c^2\,d^2\,\left (5\,a\,e^2-c\,d^2\right )}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )}{e}+\frac {4\,a^2\,c\,d\,e^4+2\,a\,c^2\,d^3\,e^2-2\,c^3\,d^5}{7\,e\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {14\,c^3\,d^4-34\,a\,c^2\,d^2\,e^2}{35\,e\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}+\frac {4\,c^3\,d^4}{7\,e\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {\left (\frac {d\,\left (\frac {56\,c^4\,d^5-72\,a\,c^3\,d^3\,e^2}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}+\frac {8\,c^4\,d^5}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )}{e}+\frac {8\,a\,c^2\,d^2\,e\,\left (8\,a\,e^2-7\,c\,d^2\right )}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {d\,\left (\frac {16\,c^5\,d^6}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {8\,c^4\,d^4\,\left (19\,a\,e^2-15\,c\,d^2\right )}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}\right )}{e}+\frac {8\,c^3\,d^3\,\left (9\,a^2\,e^4+a\,c\,d^2\,e^2-8\,c^2\,d^4\right )}{105\,e^2\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {2\,a^2\,e^3}{7\,a\,e^3-7\,c\,d^2\,e}+\frac {d\,\left (\frac {2\,c^2\,d^3}{7\,a\,e^3-7\,c\,d^2\,e}-\frac {4\,a\,c\,d\,e^2}{7\,a\,e^3-7\,c\,d^2\,e}\right )}{e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^4}-\frac {\left (\frac {28\,c^4\,d^5-36\,a\,c^3\,d^3\,e^2}{35\,e^2\,{\left (a\,e^2-c\,d^2\right )}^3}+\frac {8\,c^4\,d^5}{35\,e^2\,{\left (a\,e^2-c\,d^2\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {d\,\left (\frac {16\,c^5\,d^6}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {16\,c^4\,d^4\,\left (11\,a\,e^2-9\,c\,d^2\right )}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}\right )}{e}+\frac {16\,a\,c^3\,d^3\,\left (10\,a\,e^2-9\,c\,d^2\right )}{105\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}-\frac {\left (\frac {d\,\left (\frac {16\,c^3\,d^4-24\,a\,c^2\,d^2\,e^2}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}+\frac {4\,c^3\,d^4}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )}{e}+\frac {4\,a\,c\,d\,e\,\left (5\,a\,e^2-4\,c\,d^2\right )}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {d\,\left (\frac {8\,c^4\,d^5}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {4\,c^3\,d^3\,\left (13\,a\,e^2-9\,c\,d^2\right )}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )}{e}+\frac {24\,a^2\,c^2\,d^2\,e^4+4\,a\,c^3\,d^4\,e^2-20\,c^4\,d^6}{35\,e\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}-\frac {8\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{105\,e^2\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (d+e\,x\right )} \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)^(3/2)/(d + e*x)^6,x)

[Out]

(((d*((4*c^3*d^4)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (2*c^2*d^2*(5*a*e^2 - c*d^2))/(7*(a*e^2 - c*d^2)
*(5*a*e^3 - 5*c*d^2*e))))/e + (2*a*c^2*d^3*e^2 - 2*c^3*d^5 + 4*a^2*c*d*e^4)/(7*e*(a*e^2 - c*d^2)*(5*a*e^3 - 5*
c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^3 - (((14*c^3*d^4 - 34*a*c^2*d^2*e^2)/(35*
e*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)) + (4*c^3*d^4)/(7*e*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2
+ c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (((d*((56*c^4*d^5 - 72*a*c^3*d^3*e^2)/(35*(a*e^2 - c*d^2)^2
*(3*a*e^3 - 3*c*d^2*e)) + (8*c^4*d^5)/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e))))/e + (8*a*c^2*d^2*e*(8*a*e
^2 - 7*c*d^2))/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d
+ e*x)^2 + (((d*((16*c^5*d^6)/(105*e*(a*e^2 - c*d^2)^4) - (8*c^4*d^4*(19*a*e^2 - 15*c*d^2))/(105*e*(a*e^2 - c
*d^2)^4)))/e + (8*c^3*d^3*(9*a^2*e^4 - 8*c^2*d^4 + a*c*d^2*e^2))/(105*e^2*(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^
2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((2*a^2*e^3)/(7*a*e^3 - 7*c*d^2*e) + (d*((2*c^2*d^3)/(7*a*e^3 - 7*
c*d^2*e) - (4*a*c*d*e^2)/(7*a*e^3 - 7*c*d^2*e)))/e)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4
- (((28*c^4*d^5 - 36*a*c^3*d^3*e^2)/(35*e^2*(a*e^2 - c*d^2)^3) + (8*c^4*d^5)/(35*e^2*(a*e^2 - c*d^2)^3))*(x*(
a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((16*c^5*d^6)/(105*e*(a*e^2 - c*d^2)^4) - (16*c^4*
d^4*(11*a*e^2 - 9*c*d^2))/(105*e*(a*e^2 - c*d^2)^4)))/e + (16*a*c^3*d^3*(10*a*e^2 - 9*c*d^2))/(105*(a*e^2 - c*
d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((16*c^3*d^4 - 24*a*c^2*d^2*e^2)/(7*(
a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) + (4*c^3*d^4)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e))))/e + (4*a*c*d*e
*(5*a*e^2 - 4*c*d^2))/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)
)/(d + e*x)^3 + (((d*((8*c^4*d^5)/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (4*c^3*d^3*(13*a*e^2 - 9*c*d^
2))/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e))))/e + (4*a*c^3*d^4*e^2 - 20*c^4*d^6 + 24*a^2*c^2*d^2*e^4)/(35
*e*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (8*c
^3*d^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(105*e^2*(a*e^2 - c*d^2)^2*(d + e*x))

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