3.18.42 \(\int \frac {(d+e x)^{5/2}}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\)

Optimal. Leaf size=171 \[ \frac {16 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 c^3 d^3 \sqrt {d+e x}}+\frac {8 \sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 c^2 d^2}+\frac {2 (d+e x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d} \]

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Rubi [A]  time = 0.11, 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 {16 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 c^3 d^3 \sqrt {d+e x}}+\frac {8 \sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 c^2 d^2}+\frac {2 (d+e x)^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(c*d^2 - a*e^2)^2*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*c^3*d^3*Sqrt[d + e*x]) + (8*(c*d^2 - a*
e^2)*Sqrt[d + e*x]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(15*c^2*d^2) + (2*(d + e*x)^(3/2)*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])/(5*c*d)

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)^{5/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c d}+\frac {\left (4 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {(d+e x)^{3/2}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{5 d}\\ &=\frac {8 \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^2 d^2}+\frac {2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c d}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{15 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^3 d^3 \sqrt {d+e x}}+\frac {8 \left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 c^2 d^2}+\frac {2 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 c d}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 87, normalized size = 0.51 \begin {gather*} \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (8 a^2 e^4-4 a c d e^2 (5 d+e x)+c^2 d^2 \left (15 d^2+10 d e x+3 e^2 x^2\right )\right )}{15 c^3 d^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(8*a^2*e^4 - 4*a*c*d*e^2*(5*d + e*x) + c^2*d^2*(15*d^2 + 10*d*e*x + 3*e^2*x^2
)))/(15*c^3*d^3*Sqrt[d + e*x])

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IntegrateAlgebraic [A]  time = 0.32, size = 127, normalized size = 0.74 \begin {gather*} \frac {2 \left (8 a^2 e^4-16 a c d^2 e^2-4 a c d e^2 (d+e x)+8 c^2 d^4+4 c^2 d^3 (d+e x)+3 c^2 d^2 (d+e x)^2\right ) \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}}}{15 c^3 d^3 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(8*c^2*d^4 - 16*a*c*d^2*e^2 + 8*a^2*e^4 + 4*c^2*d^3*(d + e*x) - 4*a*c*d*e^2*(d + e*x) + 3*c^2*d^2*(d + e*x)
^2)*Sqrt[-((c*d^2*(d + e*x))/e) + a*e*(d + e*x) + (c*d*(d + e*x)^2)/e])/(15*c^3*d^3*Sqrt[d + e*x])

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fricas [A]  time = 0.42, size = 117, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} d^{2} e^{2} x^{2} + 15 \, c^{2} d^{4} - 20 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} + 2 \, {\left (5 \, 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}}{15 \, {\left (c^{3} d^{3} e x + c^{3} d^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

2/15*(3*c^2*d^2*e^2*x^2 + 15*c^2*d^4 - 20*a*c*d^2*e^2 + 8*a^2*e^4 + 2*(5*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^3*d^3*e*x + c^3*d^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

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

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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 (3 c^{2} d^{2} e^{2} x^{2}-4 a c d \,e^{3} x +10 c^{2} d^{3} e x +8 a^{2} e^{4}-20 a c \,d^{2} e^{2}+15 c^{2} d^{4}\right ) \sqrt {e x +d}}{15 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2),x)

[Out]

2/15*(c*d*x+a*e)*(3*c^2*d^2*e^2*x^2-4*a*c*d*e^3*x+10*c^2*d^3*e*x+8*a^2*e^4-20*a*c*d^2*e^2+15*c^2*d^4)*(e*x+d)^
(1/2)/c^3/d^3/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)

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maxima [A]  time = 1.24, size = 122, normalized size = 0.71 \begin {gather*} \frac {2 \, {\left (3 \, c^{3} d^{3} e^{2} x^{3} + 15 \, a c^{2} d^{4} e - 20 \, a^{2} c d^{2} e^{3} + 8 \, a^{3} e^{5} + {\left (10 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + {\left (15 \, c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} + 4 \, a^{2} c d e^{4}\right )} x\right )}}{15 \, \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)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")

[Out]

2/15*(3*c^3*d^3*e^2*x^3 + 15*a*c^2*d^4*e - 20*a^2*c*d^2*e^3 + 8*a^3*e^5 + (10*c^3*d^4*e - a*c^2*d^2*e^3)*x^2 +
(15*c^3*d^5 - 10*a*c^2*d^3*e^2 + 4*a^2*c*d*e^4)*x)/(sqrt(c*d*x + a*e)*c^3*d^3)

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mupad [B]  time = 0.92, size = 131, normalized size = 0.77 \begin {gather*} \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e\,x^2\,\sqrt {d+e\,x}}{5\,c\,d}-\frac {4\,x\,\left (2\,a\,e^2-5\,c\,d^2\right )\,\sqrt {d+e\,x}}{15\,c^2\,d^2}+\frac {\sqrt {d+e\,x}\,\left (16\,a^2\,e^4-40\,a\,c\,d^2\,e^2+30\,c^2\,d^4\right )}{15\,c^3\,d^3\,e}\right )}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(5/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*e*x^2*(d + e*x)^(1/2))/(5*c*d) - (4*x*(2*a*e^2 - 5*c*d^2)*(
d + e*x)^(1/2))/(15*c^2*d^2) + ((d + e*x)^(1/2)*(16*a^2*e^4 + 30*c^2*d^4 - 40*a*c*d^2*e^2))/(15*c^3*d^3*e)))/(
x + d/e)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral((d + e*x)**(5/2)/sqrt((d + e*x)*(a*e + c*d*x)), x)

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