3.6.83 \(\int \frac {(d+e x)^3 (f+g x)}{(d^2-e^2 x^2)^{7/2}} \, dx\) [583]

Optimal. Leaf size=117 \[ \frac {(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (2 e f-3 d g) (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(2 e f-3 d g) x}{15 d^3 e \sqrt {d^2-e^2 x^2}} \]

[Out]

1/5*(d*g+e*f)*(e*x+d)^3/d/e^2/(-e^2*x^2+d^2)^(5/2)+2/15*(-3*d*g+2*e*f)*(e*x+d)/d/e^2/(-e^2*x^2+d^2)^(3/2)+1/15
*(-3*d*g+2*e*f)*x/d^3/e/(-e^2*x^2+d^2)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {803, 667, 197} \begin {gather*} \frac {(d+e x)^3 (d g+e f)}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (d+e x) (2 e f-3 d g)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (2 e f-3 d g)}{15 d^3 e \sqrt {d^2-e^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*f + d*g)*(d + e*x)^3)/(5*d*e^2*(d^2 - e^2*x^2)^(5/2)) + (2*(2*e*f - 3*d*g)*(d + e*x))/(15*d*e^2*(d^2 - e^2
*x^2)^(3/2)) + ((2*e*f - 3*d*g)*x)/(15*d^3*e*Sqrt[d^2 - e^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 667

Int[((d_) + (e_.)*(x_))^2*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)*((a + c*x^2)^(p + 1)/(c*(p
+ 1))), x] - Dist[e^2*((p + 2)/(c*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, p}, x] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && LtQ[p, -1]

Rule 803

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g + e*f)*(
d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(p + 1))), x] - Dist[e*((m*(d*g + e*f) + 2*e*f*(p + 1))/(2*c*d*(p + 1))
), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0]
&& LtQ[p, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^3 (f+g x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {(-5 e f+3 (e f+d g)) \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d e}\\ &=\frac {(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (2 e f-3 d g) (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {(-5 e f+3 (e f+d g)) \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d e}\\ &=\frac {(e f+d g) (d+e x)^3}{5 d e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 (2 e f-3 d g) (d+e x)}{15 d e^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(2 e f-3 d g) x}{15 d^3 e \sqrt {d^2-e^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.48, size = 77, normalized size = 0.66 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (-3 d^3 g+2 e^3 f x^2-3 d e^2 x (2 f+g x)+d^2 e (7 f+9 g x)\right )}{15 d^3 e^2 (d-e x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-3*d^3*g + 2*e^3*f*x^2 - 3*d*e^2*x*(2*f + g*x) + d^2*e*(7*f + 9*g*x)))/(15*d^3*e^2*(d -
e*x)^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(408\) vs. \(2(105)=210\).
time = 0.07, size = 409, normalized size = 3.50

method	result	size
trager	\(-\frac {\left (3 d \,e^{2} g \,x^{2}-2 e^{3} f \,x^{2}-9 d^{2} e g x +6 d \,e^{2} f x +3 d^{3} g -7 d^{2} e f \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{3} \left (-e x +d \right )^{3} e^{2}}\)	\(80\)
gosper	\(-\frac {\left (e x +d \right )^{4} \left (-e x +d \right ) \left (3 d \,e^{2} g \,x^{2}-2 e^{3} f \,x^{2}-9 d^{2} e g x +6 d \,e^{2} f x +3 d^{3} g -7 d^{2} e f \right )}{15 d^{3} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\)	\(85\)
default	\(e^{3} g \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+\left (3 e^{2} d g +e^{3} f \right ) \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+\left (3 e \,d^{2} g +3 e^{2} d f \right ) \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )+\frac {d^{3} g +3 d^{2} e f}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+d^{3} f \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )\)	\(409\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(g*x+f)/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)

[Out]

e^3*g*(1/2*x^3/e^2/(-e^2*x^2+d^2)^(5/2)-3/2*d^2/e^2*(1/4*x/e^2/(-e^2*x^2+d^2)^(5/2)-1/4*d^2/e^2*(1/5*x/d^2/(-e
^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2)))))+(3*d*e^2*g+e^3*f)
*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(-e^2*x^2+d^2)^(5/2))+(3*d^2*e*g+3*d*e^2*f)*(1/4*x/e^2/(-e^2*x
^2+d^2)^(5/2)-1/4*d^2/e^2*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-
e^2*x^2+d^2)^(1/2))))+1/5*(d^3*g+3*d^2*e*f)/e^2/(-e^2*x^2+d^2)^(5/2)+d^3*f*(1/5*x/d^2/(-e^2*x^2+d^2)^(5/2)+4/5
/d^2*(1/3*x/d^2/(-e^2*x^2+d^2)^(3/2)+2/3*x/d^4/(-e^2*x^2+d^2)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (104) = 208\).
time = 0.29, size = 346, normalized size = 2.96 \begin {gather*} \frac {g x^{3} e}{2 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {3 \, d^{2} g x e^{\left (-1\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d^{3} g e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} f e^{\left (-1\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {g x e^{\left (-1\right )}}{10 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}}} + \frac {{\left (3 \, d g e^{2} + f e^{3}\right )} x^{2} e^{\left (-2\right )}}{3 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {2 \, {\left (3 \, d g e^{2} + f e^{3}\right )} d^{2} e^{\left (-4\right )}}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d f x}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {g x e^{\left (-1\right )}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{2}} + \frac {3 \, {\left (d^{2} g e + d f e^{2}\right )} x e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {4 \, f x}{15 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d} - \frac {{\left (d^{2} g e + d f e^{2}\right )} x e^{\left (-2\right )}}{5 \, {\left (-x^{2} e^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}} + \frac {8 \, f x}{15 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{3}} - \frac {2 \, {\left (d^{2} g e + d f e^{2}\right )} x e^{\left (-2\right )}}{5 \, \sqrt {-x^{2} e^{2} + d^{2}} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")

[Out]

1/2*g*x^3*e/(-x^2*e^2 + d^2)^(5/2) - 3/10*d^2*g*x*e^(-1)/(-x^2*e^2 + d^2)^(5/2) + 1/5*d^3*g*e^(-2)/(-x^2*e^2 +
d^2)^(5/2) + 3/5*d^2*f*e^(-1)/(-x^2*e^2 + d^2)^(5/2) + 1/10*g*x*e^(-1)/(-x^2*e^2 + d^2)^(3/2) + 1/3*(3*d*g*e^
2 + f*e^3)*x^2*e^(-2)/(-x^2*e^2 + d^2)^(5/2) - 2/15*(3*d*g*e^2 + f*e^3)*d^2*e^(-4)/(-x^2*e^2 + d^2)^(5/2) + 1/
5*d*f*x/(-x^2*e^2 + d^2)^(5/2) + 1/5*g*x*e^(-1)/(sqrt(-x^2*e^2 + d^2)*d^2) + 3/5*(d^2*g*e + d*f*e^2)*x*e^(-2)/
(-x^2*e^2 + d^2)^(5/2) + 4/15*f*x/((-x^2*e^2 + d^2)^(3/2)*d) - 1/5*(d^2*g*e + d*f*e^2)*x*e^(-2)/((-x^2*e^2 + d
^2)^(3/2)*d^2) + 8/15*f*x/(sqrt(-x^2*e^2 + d^2)*d^3) - 2/5*(d^2*g*e + d*f*e^2)*x*e^(-2)/(sqrt(-x^2*e^2 + d^2)*
d^4)

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Fricas [A]
time = 2.61, size = 174, normalized size = 1.49 \begin {gather*} \frac {3 \, d^{4} g + 7 \, f x^{3} e^{4} - 3 \, {\left (d g x^{3} + 7 \, d f x^{2}\right )} e^{3} + 3 \, {\left (3 \, d^{2} g x^{2} + 7 \, d^{2} f x\right )} e^{2} - {\left (9 \, d^{3} g x + 7 \, d^{3} f\right )} e + {\left (3 \, d^{3} g - 2 \, f x^{2} e^{3} + 3 \, {\left (d g x^{2} + 2 \, d f x\right )} e^{2} - {\left (9 \, d^{2} g x + 7 \, d^{2} f\right )} e\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (d^{3} x^{3} e^{5} - 3 \, d^{4} x^{2} e^{4} + 3 \, d^{5} x e^{3} - d^{6} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")

[Out]

1/15*(3*d^4*g + 7*f*x^3*e^4 - 3*(d*g*x^3 + 7*d*f*x^2)*e^3 + 3*(3*d^2*g*x^2 + 7*d^2*f*x)*e^2 - (9*d^3*g*x + 7*d
^3*f)*e + (3*d^3*g - 2*f*x^2*e^3 + 3*(d*g*x^2 + 2*d*f*x)*e^2 - (9*d^2*g*x + 7*d^2*f)*e)*sqrt(-x^2*e^2 + d^2))/
(d^3*x^3*e^5 - 3*d^4*x^2*e^4 + 3*d^5*x*e^3 - d^6*e^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(g*x+f)/(-e**2*x**2+d**2)**(7/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (104) = 208\).
time = 1.44, size = 264, normalized size = 2.26 \begin {gather*} \frac {2 \, {\left (\frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} d g e^{\left (-2\right )}}{x} - \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} d g e^{\left (-4\right )}}{x^{2}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} d g e^{\left (-6\right )}}{x^{3}} - 3 \, d g + 7 \, f e - \frac {20 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} f e^{\left (-1\right )}}{x} + \frac {40 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{2} f e^{\left (-3\right )}}{x^{2}} - \frac {30 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{3} f e^{\left (-5\right )}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )}^{4} f e^{\left (-7\right )}}{x^{4}}\right )} e^{\left (-2\right )}}{15 \, d^{3} {\left (\frac {{\left (d e + \sqrt {-x^{2} e^{2} + d^{2}} e\right )} e^{\left (-2\right )}}{x} - 1\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(g*x+f)/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")

[Out]

2/15*(15*(d*e + sqrt(-x^2*e^2 + d^2)*e)*d*g*e^(-2)/x - 15*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*d*g*e^(-4)/x^2 + 15
*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*d*g*e^(-6)/x^3 - 3*d*g + 7*f*e - 20*(d*e + sqrt(-x^2*e^2 + d^2)*e)*f*e^(-1)/
x + 40*(d*e + sqrt(-x^2*e^2 + d^2)*e)^2*f*e^(-3)/x^2 - 30*(d*e + sqrt(-x^2*e^2 + d^2)*e)^3*f*e^(-5)/x^3 + 15*(
d*e + sqrt(-x^2*e^2 + d^2)*e)^4*f*e^(-7)/x^4)*e^(-2)/(d^3*((d*e + sqrt(-x^2*e^2 + d^2)*e)*e^(-2)/x - 1)^5)

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Mupad [B]
time = 2.79, size = 79, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {d^2-e^2\,x^2}\,\left (3\,g\,d^3-9\,g\,d^2\,e\,x-7\,f\,d^2\,e+3\,g\,d\,e^2\,x^2+6\,f\,d\,e^2\,x-2\,f\,e^3\,x^2\right )}{15\,d^3\,e^2\,{\left (d-e\,x\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((d^2 - e^2*x^2)^(1/2)*(3*d^3*g - 2*e^3*f*x^2 - 7*d^2*e*f + 6*d*e^2*f*x - 9*d^2*e*g*x + 3*d*e^2*g*x^2))/(15*d
^3*e^2*(d - e*x)^3)

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