### 3.870 $$\int (d+e x)^{3/2} (c d^2-c e^2 x^2)^{3/2} \, dx$$

Optimal. Leaf size=160 $-\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}$

[Out]

(-256*d^3*(c*d^2 - c*e^2*x^2)^(5/2))/(1155*c*e*(d + e*x)^(5/2)) - (64*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(231*c*e*
(d + e*x)^(3/2)) - (8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(33*c*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^
2)^(5/2))/(11*c*e)

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Rubi [A]  time = 0.0700215, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {657, 649} $-\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-256*d^3*(c*d^2 - c*e^2*x^2)^(5/2))/(1155*c*e*(d + e*x)^(5/2)) - (64*d^2*(c*d^2 - c*e^2*x^2)^(5/2))/(231*c*e*
(d + e*x)^(3/2)) - (8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(33*c*e*Sqrt[d + e*x]) - (2*Sqrt[d + e*x]*(c*d^2 - c*e^2*x^
2)^(5/2))/(11*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)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx &=-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac{1}{11} (12 d) \int \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2} \, dx\\ &=-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac{1}{33} \left (32 d^2\right ) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx\\ &=-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}+\frac{1}{231} \left (128 d^3\right ) \int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\\ &=-\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{5/2}}{1155 c e (d+e x)^{5/2}}-\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{5/2}}{231 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{33 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{5/2}}{11 c e}\\ \end{align*}

Mathematica [A]  time = 0.0567404, size = 73, normalized size = 0.46 $-\frac{2 c (d-e x)^2 \left (755 d^2 e x+533 d^3+455 d e^2 x^2+105 e^3 x^3\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{1155 e \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(3/2)*(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)]*(533*d^3 + 755*d^2*e*x + 455*d*e^2*x^2 + 105*e^3*x^3))/(1155*e*Sqrt[
d + e*x])

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Maple [A]  time = 0.049, size = 66, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 105\,{e}^{3}{x}^{3}+455\,d{e}^{2}{x}^{2}+755\,{d}^{2}xe+533\,{d}^{3} \right ) }{1155\,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)^(3/2)*(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

-2/1155*(-e*x+d)*(105*e^3*x^3+455*d*e^2*x^2+755*d^2*e*x+533*d^3)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)

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Maxima [A]  time = 1.28505, size = 130, normalized size = 0.81 \begin{align*} -\frac{2 \,{\left (105 \, c^{\frac{3}{2}} e^{5} x^{5} + 245 \, c^{\frac{3}{2}} d e^{4} x^{4} - 50 \, c^{\frac{3}{2}} d^{2} e^{3} x^{3} - 522 \, c^{\frac{3}{2}} d^{3} e^{2} x^{2} - 311 \, c^{\frac{3}{2}} d^{4} e x + 533 \, c^{\frac{3}{2}} d^{5}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{1155 \,{\left (e^{2} x + d e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1155*(105*c^(3/2)*e^5*x^5 + 245*c^(3/2)*d*e^4*x^4 - 50*c^(3/2)*d^2*e^3*x^3 - 522*c^(3/2)*d^3*e^2*x^2 - 311*
c^(3/2)*d^4*e*x + 533*c^(3/2)*d^5)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

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Fricas [A]  time = 2.46514, size = 220, normalized size = 1.38 \begin{align*} -\frac{2 \,{\left (105 \, c e^{5} x^{5} + 245 \, c d e^{4} x^{4} - 50 \, c d^{2} e^{3} x^{3} - 522 \, c d^{3} e^{2} x^{2} - 311 \, c d^{4} e x + 533 \, c d^{5}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{1155 \,{\left (e^{2} x + d e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1155*(105*c*e^5*x^5 + 245*c*d*e^4*x^4 - 50*c*d^2*e^3*x^3 - 522*c*d^3*e^2*x^2 - 311*c*d^4*e*x + 533*c*d^5)*s
qrt(-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{3}{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(-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)**(3/2), 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{3}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(-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)^(3/2), x)