3.21.73 \(\int \frac {(d+e x)^{7/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [2073]

Optimal. Leaf size=107 \[ \frac {4 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {2 (d+e x)^{5/2}}{c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662} \begin {gather*} \frac {4 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}-\frac {2 (d+e x)^{5/2}}{c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), 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 + p, 0]

Rule 670

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))), In
t[(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] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 53, normalized size = 0.50 \begin {gather*} -\frac {2 (d+e x)^{3/2} \left (2 a e^2+c d (d+3 e x)\right )}{3 c^2 d^2 ((a e+c d x) (d+e x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

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

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Maple [A]
time = 0.75, size = 60, normalized size = 0.56

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (3 c d e x +2 e^{2} a +c \,d^{2}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right )^{2} c^{2} d^{2}}\)	\(60\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (3 c d e x +2 e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{3 c^{2} d^{2} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 52, normalized size = 0.49 \begin {gather*} -\frac {2 \, {\left (3 \, c d x e + c d^{2} + 2 \, a e^{2}\right )}}{3 \, {\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )} \sqrt {c d x + a e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

-2/3*(3*c*d*x*e + c*d^2 + 2*a*e^2)/((c^3*d^3*x + a*c^2*d^2*e)*sqrt(c*d*x + a*e))

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Fricas [A]
time = 2.71, size = 131, normalized size = 1.22 \begin {gather*} -\frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (3 \, c d x e + c d^{2} + 2 \, a e^{2}\right )} \sqrt {x e + d}}{3 \, {\left (c^{4} d^{5} x^{2} + a^{2} c^{2} d^{2} x e^{3} + {\left (2 \, a c^{3} d^{3} x^{2} + a^{2} c^{2} d^{3}\right )} e^{2} + {\left (c^{4} d^{4} x^{3} + 2 \, a c^{3} d^{4} x\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3061 deep

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Giac [A]
time = 1.16, size = 101, normalized size = 0.94 \begin {gather*} \frac {4 \, e^{2}}{3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{2}} - \frac {2 \, {\left (c d^{2} e^{3} - a e^{5} + 3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )} e^{2}\right )}}{3 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.07, size = 147, normalized size = 1.37 \begin {gather*} -\frac {\left (\frac {2\,x\,\sqrt {d+e\,x}}{c^3\,d^3}+\frac {\left (\frac {2\,c\,d^2}{3}+\frac {4\,a\,e^2}{3}\right )\,\sqrt {d+e\,x}}{c^4\,d^4\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3+\frac {a^2\,e}{c^2\,d}+\frac {a\,x\,\left (2\,c\,d^2+a\,e^2\right )}{c^2\,d^2}+\frac {x^2\,\left (c^4\,d^5+2\,a\,c^3\,d^3\,e^2\right )}{c^4\,d^4\,e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

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

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