### 3.869 $$\int (d+e x)^{5/2} (c d^2-c e^2 x^2)^{3/2} \, dx$$

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

[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)

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Rubi [A]  time = 0.0996525, antiderivative size = 201, normalized size of antiderivative = 1., 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} $-\frac{4096 d^4 \left (c d^2-c e^2 x^2\right )^{5/2}}{15015 c e (d+e x)^{5/2}}-\frac{1024 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{3003 c e (d+e x)^{3/2}}-\frac{128 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{429 c e \sqrt{d+e x}}-\frac{32 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{143 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{5/2}}{13 c e}$

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 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]

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]

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 = 0.0673561, size = 84, normalized size = 0.42 $-\frac{2 c (d-e x)^2 \left (14210 d^2 e^2 x^2+16700 d^3 e x+9683 d^4+6300 d e^3 x^3+1155 e^4 x^4\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{15015 e \sqrt{d+e x}}$

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])

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Maple [A]  time = 0.043, size = 77, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 1155\,{e}^{4}{x}^{4}+6300\,d{e}^{3}{x}^{3}+14210\,{d}^{2}{e}^{2}{x}^{2}+16700\,{d}^{3}xe+9683\,{d}^{4} \right ) }{15015\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

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)

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Maxima [A]  time = 1.27707, size = 149, normalized size = 0.74 \begin{align*} -\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{align*}

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)

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Fricas [A]  time = 2.34063, size = 259, normalized size = 1.29 \begin{align*} -\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{align*}

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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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)