3.8.10 \(\int (d+e x)^{5/2} (c d^2-c e^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=201 \[ -\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}-\frac {4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {657, 649} \begin {gather*} -\frac {4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-4096*d^4*(c*d^2 - c*e^2*x^2)^(5/2))/(15015*c*e*(d + e*x)^(5/2)) - (1024*d^3*(c*d^2 - c*e^2*x^2)^(5/2))/(3003
*c*e*(d + e*x)^(3/2)) - (128*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(429*c*e*Sqrt[d + e*x]) - (32*d*Sqrt[d + e*x]*(c*d
^2 - c*e^2*x^2)^(5/2))/(143*c*e) - (2*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(5/2))/(13*c*e)

Rule 649

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p,
0]

Rule 657

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*(a + c*x^2)^(p
+ 1))/(c*(m + 2*p + 1)), x] + Dist[(2*c*d*Simplify[m + p])/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^
2)^p, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p]
, 0]

Rubi steps

\begin {align*} \int (d+e x)^{5/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx &=-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac {1}{13} (16 d) \int (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\\ &=-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac {1}{143} \left (192 d^2\right ) \int \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\\ &=-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac {1}{429} \left (512 d^3\right ) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx\\ &=-\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}+\frac {\left (2048 d^4\right ) \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{3003}\\ &=-\frac {4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac {1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac {128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt {d+e x}}-\frac {32 d \sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac {2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 84, normalized size = 0.42 \begin {gather*} -\frac {2 c (d-e x)^2 \left (9683 d^4+16700 d^3 e x+14210 d^2 e^2 x^2+6300 d e^3 x^3+1155 e^4 x^4\right ) \sqrt {c \left (d^2-e^2 x^2\right )}}{15015 e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)*(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*c*(d - e*x)^2*Sqrt[c*(d^2 - e^2*x^2)]*(9683*d^4 + 16700*d^3*e*x + 14210*d^2*e^2*x^2 + 6300*d*e^3*x^3 + 115
5*e^4*x^4))/(15015*e*Sqrt[d + e*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.58, size = 90, normalized size = 0.45 \begin {gather*} -\frac {2 \left (2048 d^4+2560 d^3 (d+e x)+2240 d^2 (d+e x)^2+1680 d (d+e x)^3+1155 (d+e x)^4\right ) \left (2 c d (d+e x)-c (d+e x)^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(5/2)*(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*(2*c*d*(d + e*x) - c*(d + e*x)^2)^(5/2)*(2048*d^4 + 2560*d^3*(d + e*x) + 2240*d^2*(d + e*x)^2 + 1680*d*(d
+ e*x)^3 + 1155*(d + e*x)^4))/(15015*c*e*(d + e*x)^(5/2))

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 107, normalized size = 0.53 \begin {gather*} -\frac {2 \, {\left (1155 \, c e^{6} x^{6} + 3990 \, c d e^{5} x^{5} + 2765 \, c d^{2} e^{4} x^{4} - 5420 \, c d^{3} e^{3} x^{3} - 9507 \, c d^{4} e^{2} x^{2} - 2666 \, c d^{5} e x + 9683 \, c d^{6}\right )} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d}}{15015 \, {\left (e^{2} x + d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2/15015*(1155*c*e^6*x^6 + 3990*c*d*e^5*x^5 + 2765*c*d^2*e^4*x^4 - 5420*c*d^3*e^3*x^3 - 9507*c*d^4*e^2*x^2 - 2
666*c*d^5*e*x + 9683*c*d^6)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(e^2*x + d*e)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}} {\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)*(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="giac")

[Out]

integrate((-c*e^2*x^2 + c*d^2)^(3/2)*(e*x + d)^(5/2), x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 77, normalized size = 0.38 \begin {gather*} -\frac {2 \left (-e x +d \right ) \left (1155 e^{4} x^{4}+6300 e^{3} x^{3} d +14210 d^{2} e^{2} x^{2}+16700 d^{3} x e +9683 d^{4}\right ) \left (-c \,e^{2} x^{2}+c \,d^{2}\right )^{\frac {3}{2}}}{15015 \left (e x +d \right )^{\frac {3}{2}} e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

-2/15015*(-e*x+d)*(1155*e^4*x^4+6300*d*e^3*x^3+14210*d^2*e^2*x^2+16700*d^3*e*x+9683*d^4)*(-c*e^2*x^2+c*d^2)^(3
/2)/e/(e*x+d)^(3/2)

________________________________________________________________________________________

maxima [A]  time = 1.62, size = 110, normalized size = 0.55 \begin {gather*} -\frac {2 \, {\left (1155 \, c^{\frac {3}{2}} e^{6} x^{6} + 3990 \, c^{\frac {3}{2}} d e^{5} x^{5} + 2765 \, c^{\frac {3}{2}} d^{2} e^{4} x^{4} - 5420 \, c^{\frac {3}{2}} d^{3} e^{3} x^{3} - 9507 \, c^{\frac {3}{2}} d^{4} e^{2} x^{2} - 2666 \, c^{\frac {3}{2}} d^{5} e x + 9683 \, c^{\frac {3}{2}} d^{6}\right )} {\left (e x + d\right )} \sqrt {-e x + d}}{15015 \, {\left (e^{2} x + d e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(-c*e^2*x^2+c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/15015*(1155*c^(3/2)*e^6*x^6 + 3990*c^(3/2)*d*e^5*x^5 + 2765*c^(3/2)*d^2*e^4*x^4 - 5420*c^(3/2)*d^3*e^3*x^3
- 9507*c^(3/2)*d^4*e^2*x^2 - 2666*c^(3/2)*d^5*e*x + 9683*c^(3/2)*d^6)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

________________________________________________________________________________________

mupad [B]  time = 0.70, size = 116, normalized size = 0.58 \begin {gather*} -\frac {16384\,c\,d^6\,\sqrt {c\,d^2-c\,e^2\,x^2}}{15015\,e\,\sqrt {d+e\,x}}-\frac {2\,c\,\sqrt {c\,d^2-c\,e^2\,x^2}\,\sqrt {d+e\,x}\,\left (1491\,d^5-4157\,d^4\,e\,x-5350\,d^3\,e^2\,x^2-70\,d^2\,e^3\,x^3+2835\,d\,e^4\,x^4+1155\,e^5\,x^5\right )}{15015\,e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d^2 - c*e^2*x^2)^(3/2)*(d + e*x)^(5/2),x)

[Out]

- (16384*c*d^6*(c*d^2 - c*e^2*x^2)^(1/2))/(15015*e*(d + e*x)^(1/2)) - (2*c*(c*d^2 - c*e^2*x^2)^(1/2)*(d + e*x)
^(1/2)*(1491*d^5 + 1155*e^5*x^5 + 2835*d*e^4*x^4 - 5350*d^3*e^2*x^2 - 70*d^2*e^3*x^3 - 4157*d^4*e*x))/(15015*e
)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Integral((-c*(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)**(5/2), x)

________________________________________________________________________________________