3.9.87 \(\int \frac {(d+e x)^{7/2}}{(c d^2-c e^2 x^2)^{3/2}} \, dx\) [887]

Optimal. Leaf size=119 \[ \frac {64 d^2 \sqrt {d+e x}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}} \]

[Out]

-16/3*d*(e*x+d)^(3/2)/c/e/(-c*e^2*x^2+c*d^2)^(1/2)-2/3*(e*x+d)^(5/2)/c/e/(-c*e^2*x^2+c*d^2)^(1/2)+64/3*d^2*(e*
x+d)^(1/2)/c/e/(-c*e^2*x^2+c*d^2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {671, 663} \begin {gather*} -\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {64 d^2 \sqrt {d+e x}}{3 c e \sqrt {c d^2-c e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(64*d^2*Sqrt[d + e*x])/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (16*d*(d + e*x)^(3/2))/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2]
) - (2*(d + e*x)^(5/2))/(3*c*e*Sqrt[c*d^2 - c*e^2*x^2])

Rule 663

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 671

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 \frac {(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{3} (8 d) \int \frac {(d+e x)^{5/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}+\frac {1}{3} \left (32 d^2\right ) \int \frac {(d+e x)^{3/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {64 d^2 \sqrt {d+e x}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {16 d (d+e x)^{3/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}-\frac {2 (d+e x)^{5/2}}{3 c e \sqrt {c d^2-c e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 55, normalized size = 0.46 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (-23 d^2+10 d e x+e^2 x^2\right )}{3 c e \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-23*d^2 + 10*d*e*x + e^2*x^2))/(3*c*e*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]
time = 0.48, size = 59, normalized size = 0.50

method	result	size
gosper	\(\frac {2 \left (-e x +d \right ) \left (-e^{2} x^{2}-10 d x e +23 d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3 e \left (-x^{2} c \,e^{2}+c \,d^{2}\right )^{\frac {3}{2}}}\)	\(55\)
default	\(\frac {2 \sqrt {c \left (-e^{2} x^{2}+d^{2}\right )}\, \left (-e^{2} x^{2}-10 d x e +23 d^{2}\right )}{3 \sqrt {e x +d}\, c^{2} \left (-e x +d \right ) e}\)	\(59\)
risch	\(\frac {2 \left (e x +11 d \right ) \left (-e x +d \right ) \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{3 e \sqrt {-c \left (e x -d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, c}+\frac {8 d^{2} \sqrt {-\frac {c \left (e^{2} x^{2}-d^{2}\right )}{e x +d}}\, \sqrt {e x +d}}{e \sqrt {c \left (-e x +d \right )}\, \sqrt {-c \left (e^{2} x^{2}-d^{2}\right )}\, c}\)	\(156\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3/(e*x+d)^(1/2)*(c*(-e^2*x^2+d^2))^(1/2)/c^2*(-e^2*x^2-10*d*e*x+23*d^2)/(-e*x+d)/e

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Maxima [A]
time = 0.29, size = 43, normalized size = 0.36 \begin {gather*} -\frac {2 \, {\left (\sqrt {c} x^{2} e^{2} + 10 \, \sqrt {c} d x e - 23 \, \sqrt {c} d^{2}\right )} e^{\left (-1\right )}}{3 \, \sqrt {-x e + d} c^{2}} \end {gather*}

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)^(3/2),x, algorithm="maxima")

[Out]

-2/3*(sqrt(c)*x^2*e^2 + 10*sqrt(c)*d*x*e - 23*sqrt(c)*d^2)*e^(-1)/(sqrt(-x*e + d)*c^2)

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Fricas [A]
time = 3.31, size = 66, normalized size = 0.55 \begin {gather*} \frac {2 \, \sqrt {-c x^{2} e^{2} + c d^{2}} {\left (x^{2} e^{2} + 10 \, d x e - 23 \, d^{2}\right )} \sqrt {x e + d}}{3 \, {\left (c^{2} x^{2} e^{3} - c^{2} d^{2} e\right )}} \end {gather*}

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)^(3/2),x, algorithm="fricas")

[Out]

2/3*sqrt(-c*x^2*e^2 + c*d^2)*(x^2*e^2 + 10*d*x*e - 23*d^2)*sqrt(x*e + d)/(c^2*x^2*e^3 - c^2*d^2*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

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)**(3/2),x)

[Out]

Integral((d + e*x)**(7/2)/(-c*(-d + e*x)*(d + e*x))**(3/2), x)

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Giac [A]
time = 1.41, size = 100, normalized size = 0.84 \begin {gather*} -\frac {32 \, \sqrt {2} d^{2} e^{\left (-1\right )}}{3 \, \sqrt {c d} c} + \frac {8 \, d^{2} e^{\left (-1\right )}}{\sqrt {-{\left (x e + d\right )} c + 2 \, c d} c} + \frac {2 \, {\left (12 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} c^{7} d e^{2} - {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c^{6} e^{2}\right )} e^{\left (-3\right )}}{3 \, c^{9}} \end {gather*}

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)^(3/2),x, algorithm="giac")

[Out]

-32/3*sqrt(2)*d^2*e^(-1)/(sqrt(c*d)*c) + 8*d^2*e^(-1)/(sqrt(-(x*e + d)*c + 2*c*d)*c) + 2/3*(12*sqrt(-(x*e + d)
*c + 2*c*d)*c^7*d*e^2 - (-(x*e + d)*c + 2*c*d)^(3/2)*c^6*e^2)*e^(-3)/c^9

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Mupad [B]
time = 0.67, size = 86, normalized size = 0.72 \begin {gather*} \frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {2\,x^2\,\sqrt {d+e\,x}}{3\,c^2\,e}-\frac {46\,d^2\,\sqrt {d+e\,x}}{3\,c^2\,e^3}+\frac {20\,d\,x\,\sqrt {d+e\,x}}{3\,c^2\,e^2}\right )}{x^2-\frac {d^2}{e^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x)

[Out]

((c*d^2 - c*e^2*x^2)^(1/2)*((2*x^2*(d + e*x)^(1/2))/(3*c^2*e) - (46*d^2*(d + e*x)^(1/2))/(3*c^2*e^3) + (20*d*x
*(d + e*x)^(1/2))/(3*c^2*e^2)))/(x^2 - d^2/e^2)

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