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

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

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Rubi [A]  time = 0.08, antiderivative size = 160, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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 {gather*}

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 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 \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*}

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Mathematica [A]  time = 0.07, size = 70, normalized size = 0.44 \begin {gather*} -\frac {2 (d-e x) \sqrt {d+e x} \left (177 d^3+71 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )}{35 e \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.15, size = 78, normalized size = 0.49 \begin {gather*} -\frac {2 \left (128 d^3+32 d^2 (d+e x)+12 d (d+e x)^2+5 (d+e x)^3\right ) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{35 c e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2*c*d*(d + e*x) - c*(d + e*x)^2]*(128*d^3 + 32*d^2*(d + e*x) + 12*d*(d + e*x)^2 + 5*(d + e*x)^3))/(35
*c*e*Sqrt[d + e*x])

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fricas [A]  time = 0.40, size = 69, normalized size = 0.43 \begin {gather*} -\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 {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)^(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {-c e^{2} x^{2} + c d^{2}}}\,{d x} \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)^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 66, normalized size = 0.41 \begin {gather*} -\frac {2 \left (-e x +d \right ) \left (5 e^{3} x^{3}+27 e^{2} x^{2} d +71 d^{2} x e +177 d^{3}\right ) \sqrt {e x +d}}{35 \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}\, e} \end {gather*}

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 = 1.47, size = 57, normalized size = 0.36 \begin {gather*} \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 {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)^(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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mupad [B]  time = 0.68, size = 98, normalized size = 0.61 \begin {gather*} -\frac {\sqrt {c\,d^2-c\,e^2\,x^2}\,\left (\frac {354\,d^3\,\sqrt {d+e\,x}}{35\,c\,e^2}+\frac {54\,d\,x^2\,\sqrt {d+e\,x}}{35\,c}+\frac {2\,e\,x^3\,\sqrt {d+e\,x}}{7\,c}+\frac {142\,d^2\,x\,\sqrt {d+e\,x}}{35\,c\,e}\right )}{x+\frac {d}{e}} \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)^(1/2),x)

[Out]

-((c*d^2 - c*e^2*x^2)^(1/2)*((354*d^3*(d + e*x)^(1/2))/(35*c*e^2) + (54*d*x^2*(d + e*x)^(1/2))/(35*c) + (2*e*x
^3*(d + e*x)^(1/2))/(7*c) + (142*d^2*x*(d + e*x)^(1/2))/(35*c*e)))/(x + d/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}}}{\sqrt {- c \left (- d + e x\right ) \left (d + e x\right )}}\, 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)**(1/2),x)

[Out]

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

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