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

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

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Rubi [A]  time = 0.12, 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 = {656, 648} \begin {gather*} \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 {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 {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 648

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 656

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 (e x-3 d)+c^2 d^2 \left (3 d^2-6 d e x-e^2 x^2\right )\right )}{3 c^3 d^3 \sqrt {(d+e x) (a e+c d 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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IntegrateAlgebraic [A]  time = 1.21, size = 115, normalized size = 0.67 \begin {gather*} \frac {2 (d+e x)^{3/2} (a e+c d x) \left (-3 a^2 e^4+6 a c d^2 e^2+6 c d^2 e (a e+c d x)-6 a e^3 (a e+c d x)+e^2 (a e+c d x)^2-3 c^2 d^4\right )}{3 c^3 d^3 ((d+e x) (a e+c d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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*(a*e + c*d*x)*(d + e*x)^(3/2)*(-3*c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4 + 6*c*d^2*e*(a*e + c*d*x) - 6*a*e^3*(
a*e + c*d*x) + e^2*(a*e + c*d*x)^2))/(3*c^3*d^3*((a*e + c*d*x)*(d + e*x))^(3/2))

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fricas [A]  time = 0.40, size = 141, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} e^{2} x^{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 )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3 \, {\left (c^{4} d^{4} e x^{2} + a c^{3} d^{4} e + {\left (c^{4} d^{5} + a c^{3} d^{3} e^{2}\right )} x\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*(c^2*d^2*e^2*x^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*e*x^
2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^4*d^4*e*x^2 + a*c^3*d^4*e + (c^4*d^5 + a*c^3*d^3*e^2)*x)

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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((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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.81Unable to transpose Error: Bad Argument Value

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maple [A]  time = 0.05, size = 110, normalized size = 0.64 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-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}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.25, size = 79, normalized size = 0.46 \begin {gather*} \frac {2 \, {\left (c^{2} d^{2} e^{2} x^{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*e^2*x^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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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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sympy [F(-1)]  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((e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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