3.21.65 \(\int \frac {(d+e x)^{7/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [2065]

Optimal. Leaf size=171 \[ -\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {8 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]

[Out]

8/3*(-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)^(1/2)+2/3*(e*x+d)^(5/2)/c/d/(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-16/3*(-a*e^2+c*d^2)^2*(e*x+d)^(1/2)/c^3/d^3/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {16 \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{3 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {8 (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^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)^(3/2),x]

[Out]

(-16*(c*d^2 - a*e^2)^2*Sqrt[d + e*x])/(3*c^3*d^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (8*(c*d^2 - a*
e^2)*(d + e*x)^(3/2))/(3*c^2*d^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (2*(d + e*x)^(5/2))/(3*c*d*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^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 )^{3/2}} \, dx &=\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (4 \left (d^2-\frac {a e^2}{c}\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 )^{3/2}} \, dx}{3 d}\\ &=\frac {8 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (8 \left (c d^2-a e^2\right )^2\right ) \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c^2 d^2}\\ &=-\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{3 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {8 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 (d+e x)^{5/2}}{3 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 87, normalized size = 0.51 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (8 a^2 e^4+4 a c d e^2 (-3 d+e x)+c^2 d^2 \left (3 d^2-6 d e x-e^2 x^2\right )\right )}{3 c^3 d^3 \sqrt {(a e+c d x) (d+e x)}} \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)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(8*a^2*e^4 + 4*a*c*d*e^2*(-3*d + e*x) + c^2*d^2*(3*d^2 - 6*d*e*x - e^2*x^2)))/(3*c^3*d^3*Sqr
t[(a*e + c*d*x)*(d + e*x)])

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Maple [A]
time = 0.86, size = 102, normalized size = 0.60

method	result	size
default	\(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-e^{2} x^{2} c^{2} d^{2}+4 a c d \,e^{3} x -6 c^{2} d^{3} e x +8 a^{2} e^{4}-12 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right )}{3 \sqrt {e x +d}\, \left (c d x +a e \right ) c^{3} d^{3}}\)	\(102\)
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (-e^{2} x^{2} c^{2} d^{2}+4 a c d \,e^{3} x -6 c^{2} d^{3} e x +8 a^{2} e^{4}-12 a c \,d^{2} e^{2}+3 c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 c^{3} d^{3} \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}}\)	\(110\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/3/(e*x+d)^(1/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(-c^2*d^2*e^2*x^2+4*a*c*d*e^3*x-6*c^2*d^3*e*x+8*a^2*e^4-12*a*c*
d^2*e^2+3*c^2*d^4)/(c*d*x+a*e)/c^3/d^3

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Maxima [A]
time = 0.30, size = 77, normalized size = 0.45 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} x^{2} e^{2} - 3 \, c^{2} d^{4} + 12 \, a c d^{2} e^{2} - 8 \, a^{2} e^{4} + 2 \, {\left (3 \, c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x\right )}}{3 \, \sqrt {c d x + a e} c^{3} d^{3}} \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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.74, size = 137, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (6 \, c^{2} d^{3} x e - 3 \, c^{2} d^{4} - 4 \, a c d x e^{3} - 8 \, a^{2} e^{4} + {\left (c^{2} d^{2} x^{2} + 12 \, a c d^{2}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {x e + d}}{3 \, {\left (c^{4} d^{5} x + a c^{3} d^{3} x e^{2} + {\left (c^{4} d^{4} x^{2} + a c^{3} d^{4}\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)^(3/2),x, algorithm="fricas")

[Out]

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

[Out]

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

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Giac [A]
time = 1.41, size = 227, normalized size = 1.33 \begin {gather*} -\frac {2 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}}{\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{3} d^{3}} + \frac {16 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}}{3 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{3}} + \frac {2 \, {\left (6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{8} e^{3} - 6 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{6} e^{5} + {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{6} e^{2}\right )} e^{\left (-3\right )}}{3 \, c^{9} d^{9}} \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)^(3/2),x, algorithm="giac")

[Out]

-2*(c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)/(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^3*d^3) + 16/3*(c^2*d^4*e -
2*a*c*d^2*e^3 + a^2*e^5)/(sqrt(-c*d^2*e + a*e^3)*c^3*d^3) + 2/3*(6*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^
7*d^8*e^3 - 6*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^6*d^6*e^5 + ((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2
)*c^6*d^6*e^2)*e^(-3)/(c^9*d^9)

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Mupad [B]
time = 1.10, size = 174, normalized size = 1.02 \begin {gather*} \frac {\left (\frac {2\,e\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,d^2}-\frac {\sqrt {d+e\,x}\,\left (16\,a^2\,e^4-24\,a\,c\,d^2\,e^2+6\,c^2\,d^4\right )}{3\,c^4\,d^4\,e}+\frac {x\,\left (12\,c^2\,d^3\,e-8\,a\,c\,d\,e^3\right )\,\sqrt {d+e\,x}}{3\,c^4\,d^4\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\frac {a}{c}+x^2+\frac {x\,\left (3\,c^4\,d^5+3\,a\,c^3\,d^3\,e^2\right )}{3\,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)^(3/2),x)

[Out]

(((2*e*x^2*(d + e*x)^(1/2))/(3*c^2*d^2) - ((d + e*x)^(1/2)*(16*a^2*e^4 + 6*c^2*d^4 - 24*a*c*d^2*e^2))/(3*c^4*d
^4*e) + (x*(12*c^2*d^3*e - 8*a*c*d*e^3)*(d + e*x)^(1/2))/(3*c^4*d^4*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2
)^(1/2))/(a/c + x^2 + (x*(3*c^4*d^5 + 3*a*c^3*d^3*e^2))/(3*c^4*d^4*e))

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