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

Optimal. Leaf size=233 \[ \frac {32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt {d+e x}}+\frac {4 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d} \]

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Rubi [A]  time = 0.19, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt {d+e x}}+\frac {4 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 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]

(32*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(315*c^4*d^4*(d + e*x)^(3/2)) + (16*(c*d^
2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(105*c^3*d^3*Sqrt[d + e*x]) + (4*(c*d^2 - a*e^2)*S
qrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(21*c^2*d^2) + (2*(d + e*x)^(3/2)*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(3/2))/(9*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 (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} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 c d}+\frac {\left (2 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{3 d}\\ &=\frac {4 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac {2 (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}}{9 c d}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{21 d^2}\\ &=\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt {d+e x}}+\frac {4 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac {2 (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}}{9 c d}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{105 d^3}\\ &=\frac {32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt {d+e x}}+\frac {4 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac {2 (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}}{9 c d}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 131, normalized size = 0.56 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (-16 a^3 e^6+24 a^2 c d e^4 (3 d+e x)-6 a c^2 d^2 e^2 \left (21 d^2+18 d e x+5 e^2 x^2\right )+c^3 d^3 \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )\right )}{315 c^4 d^4 (d+e x)^{3/2}} \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*((a*e + c*d*x)*(d + e*x))^(3/2)*(-16*a^3*e^6 + 24*a^2*c*d*e^4*(3*d + e*x) - 6*a*c^2*d^2*e^2*(21*d^2 + 18*d*
e*x + 5*e^2*x^2) + c^3*d^3*(105*d^3 + 189*d^2*e*x + 135*d*e^2*x^2 + 35*e^3*x^3)))/(315*c^4*d^4*(d + e*x)^(3/2)
)

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IntegrateAlgebraic [A]  time = 0.49, size = 285, normalized size = 1.22 \begin {gather*} \frac {2 \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}} \left (-16 a^4 e^8+64 a^3 c d^2 e^6+8 a^3 c d e^6 (d+e x)-96 a^2 c^2 d^4 e^4-24 a^2 c^2 d^3 e^4 (d+e x)-6 a^2 c^2 d^2 e^4 (d+e x)^2+64 a c^3 d^6 e^2+24 a c^3 d^5 e^2 (d+e x)+12 a c^3 d^4 e^2 (d+e x)^2+5 a c^3 d^3 e^2 (d+e x)^3-16 c^4 d^8-8 c^4 d^7 (d+e x)-6 c^4 d^6 (d+e x)^2-5 c^4 d^5 (d+e x)^3+35 c^4 d^4 (d+e x)^4\right )}{315 c^4 d^4 e \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*Sqrt[-((c*d^2*(d + e*x))/e) + a*e*(d + e*x) + (c*d*(d + e*x)^2)/e]*(-16*c^4*d^8 + 64*a*c^3*d^6*e^2 - 96*a^2
*c^2*d^4*e^4 + 64*a^3*c*d^2*e^6 - 16*a^4*e^8 - 8*c^4*d^7*(d + e*x) + 24*a*c^3*d^5*e^2*(d + e*x) - 24*a^2*c^2*d
^3*e^4*(d + e*x) + 8*a^3*c*d*e^6*(d + e*x) - 6*c^4*d^6*(d + e*x)^2 + 12*a*c^3*d^4*e^2*(d + e*x)^2 - 6*a^2*c^2*
d^2*e^4*(d + e*x)^2 - 5*c^4*d^5*(d + e*x)^3 + 5*a*c^3*d^3*e^2*(d + e*x)^3 + 35*c^4*d^4*(d + e*x)^4))/(315*c^4*
d^4*e*Sqrt[d + e*x])

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fricas [A]  time = 0.40, size = 230, normalized size = 0.99 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \, {\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\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}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\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]

2/315*(35*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 126*a^2*c^2*d^4*e^3 + 72*a^3*c*d^2*e^5 - 16*a^4*e^7 + 5*(27*c^4*
d^5*e^2 + a*c^3*d^3*e^4)*x^3 + 3*(63*c^4*d^6*e + 9*a*c^3*d^4*e^3 - 2*a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 + 63*
a*c^3*d^5*e^2 - 36*a^2*c^2*d^3*e^4 + 8*a^3*c*d*e^6)*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 + c^4*d^5)

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

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maple [A]  time = 0.06, size = 168, normalized size = 0.72 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-35 c^{3} d^{3} e^{3} x^{3}+30 a \,c^{2} d^{2} e^{4} x^{2}-135 c^{3} d^{4} e^{2} x^{2}-24 a^{2} c d \,e^{5} x +108 a \,c^{2} d^{3} e^{3} x -189 c^{3} d^{5} e x +16 a^{3} e^{6}-72 a^{2} c \,d^{2} e^{4}+126 a \,c^{2} d^{4} e^{2}-105 c^{3} d^{6}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 \sqrt {e x +d}\, c^{4} d^{4}} \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]

-2/315*(c*d*x+a*e)*(-35*c^3*d^3*e^3*x^3+30*a*c^2*d^2*e^4*x^2-135*c^3*d^4*e^2*x^2-24*a^2*c*d*e^5*x+108*a*c^2*d^
3*e^3*x-189*c^3*d^5*e*x+16*a^3*e^6-72*a^2*c*d^2*e^4+126*a*c^2*d^4*e^2-105*c^3*d^6)*(c*d*e*x^2+a*e^2*x+c*d^2*x+
a*d*e)^(1/2)/c^4/d^4/(e*x+d)^(1/2)

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maxima [A]  time = 1.37, size = 211, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \, {\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\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="maxima")

[Out]

2/315*(35*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 126*a^2*c^2*d^4*e^3 + 72*a^3*c*d^2*e^5 - 16*a^4*e^7 + 5*(27*c^4*
d^5*e^2 + a*c^3*d^3*e^4)*x^3 + 3*(63*c^4*d^6*e + 9*a*c^3*d^4*e^3 - 2*a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 + 63*
a*c^3*d^5*e^2 - 36*a^2*c^2*d^3*e^4 + 8*a^3*c*d*e^6)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^4*d^4*e*x + c^4*d^5)

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mupad [B]  time = 1.05, size = 256, normalized size = 1.10 \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^2\,x^4\,\sqrt {d+e\,x}}{9}-\frac {\sqrt {d+e\,x}\,\left (32\,a^4\,e^7-144\,a^3\,c\,d^2\,e^5+252\,a^2\,c^2\,d^4\,e^3-210\,a\,c^3\,d^6\,e\right )}{315\,c^4\,d^4\,e}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (-2\,a^2\,e^4+9\,a\,c\,d^2\,e^2+63\,c^2\,d^4\right )}{105\,c^2\,d^2}+\frac {x\,\sqrt {d+e\,x}\,\left (16\,a^3\,c\,d\,e^6-72\,a^2\,c^2\,d^3\,e^4+126\,a\,c^3\,d^5\,e^2+210\,c^4\,d^7\right )}{315\,c^4\,d^4\,e}+\frac {2\,e\,x^3\,\left (27\,c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}}{63\,c\,d}\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^2*x^4*(d + e*x)^(1/2))/9 - ((d + e*x)^(1/2)*(32*a^4*e^7 -
144*a^3*c*d^2*e^5 + 252*a^2*c^2*d^4*e^3 - 210*a*c^3*d^6*e))/(315*c^4*d^4*e) + (2*x^2*(d + e*x)^(1/2)*(63*c^2*
d^4 - 2*a^2*e^4 + 9*a*c*d^2*e^2))/(105*c^2*d^2) + (x*(d + e*x)^(1/2)*(210*c^4*d^7 + 126*a*c^3*d^5*e^2 - 72*a^2
*c^2*d^3*e^4 + 16*a^3*c*d*e^6))/(315*c^4*d^4*e) + (2*e*x^3*(a*e^2 + 27*c*d^2)*(d + e*x)^(1/2))/(63*c*d)))/(x +
d/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)**(5/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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