### 3.879 $$\int \frac{(d+e x)^{7/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx$$

Optimal. Leaf size=160 $-\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}}-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}$

[Out]

(-256*d^3*Sqrt[c*d^2 - c*e^2*x^2])/(35*c*e*Sqrt[d + e*x]) - (64*d^2*Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2])/(35
*c*e) - (24*d*(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2])/(35*c*e) - (2*(d + e*x)^(5/2)*Sqrt[c*d^2 - c*e^2*x^2])/
(7*c*e)

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Rubi [A]  time = 0.0800169, 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} $-\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}}-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(7/2)/Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-256*d^3*Sqrt[c*d^2 - c*e^2*x^2])/(35*c*e*Sqrt[d + e*x]) - (64*d^2*Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2])/(35
*c*e) - (24*d*(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2])/(35*c*e) - (2*(d + e*x)^(5/2)*Sqrt[c*d^2 - c*e^2*x^2])/
(7*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 \frac{(d+e x)^{7/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx &=-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}+\frac{1}{7} (12 d) \int \frac{(d+e x)^{5/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}+\frac{1}{35} \left (96 d^2\right ) \int \frac{(d+e x)^{3/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}+\frac{1}{35} \left (128 d^3\right ) \int \frac{\sqrt{d+e x}}{\sqrt{c d^2-c e^2 x^2}} \, dx\\ &=-\frac{256 d^3 \sqrt{c d^2-c e^2 x^2}}{35 c e \sqrt{d+e x}}-\frac{64 d^2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{24 d (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{35 c e}-\frac{2 (d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}}{7 c e}\\ \end{align*}

Mathematica [A]  time = 0.0753387, size = 70, normalized size = 0.44 $-\frac{2 (d-e x) \sqrt{d+e x} \left (71 d^2 e x+177 d^3+27 d e^2 x^2+5 e^3 x^3\right )}{35 e \sqrt{c \left (d^2-e^2 x^2\right )}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(7/2)/Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-2*(d - e*x)*Sqrt[d + e*x]*(177*d^3 + 71*d^2*e*x + 27*d*e^2*x^2 + 5*e^3*x^3))/(35*e*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.043, size = 66, normalized size = 0.4 \begin{align*} -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 5\,{e}^{3}{x}^{3}+27\,d{e}^{2}{x}^{2}+71\,{d}^{2}xe+177\,{d}^{3} \right ) }{35\,e}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(1/2),x)

[Out]

-2/35*(-e*x+d)*(5*e^3*x^3+27*d*e^2*x^2+71*d^2*e*x+177*d^3)*(e*x+d)^(1/2)/e/(-c*e^2*x^2+c*d^2)^(1/2)

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Maxima [A]  time = 1.06837, size = 77, normalized size = 0.48 \begin{align*} \frac{2 \,{\left (5 \, e^{4} x^{4} + 22 \, d e^{3} x^{3} + 44 \, d^{2} e^{2} x^{2} + 106 \, d^{3} e x - 177 \, d^{4}\right )}}{35 \, \sqrt{-e x + d} \sqrt{c} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2/35*(5*e^4*x^4 + 22*d*e^3*x^3 + 44*d^2*e^2*x^2 + 106*d^3*e*x - 177*d^4)/(sqrt(-e*x + d)*sqrt(c)*e)

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Fricas [A]  time = 2.11261, size = 154, normalized size = 0.96 \begin{align*} -\frac{2 \,{\left (5 \, e^{3} x^{3} + 27 \, d e^{2} x^{2} + 71 \, d^{2} e x + 177 \, d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{35 \,{\left (c e^{2} x + c d e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

-2/35*(5*e^3*x^3 + 27*d*e^2*x^2 + 71*d^2*e*x + 177*d^3)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/(c*e^2*x + c*d*
e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{\frac{7}{2}}}{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

Integral((d + e*x)**(7/2)/sqrt(-c*(-d + e*x)*(d + e*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{\sqrt{-c e^{2} x^{2} + c d^{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(7/2)/sqrt(-c*e^2*x^2 + c*d^2), x)