3.2279 \(\int \frac{(d+e x)^{7/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=217 \[ \frac{2 (d+e x)^{3/2} (-4 b e g+7 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{4 \sqrt{d+e x} (-4 b e g+7 c d g+c e f)}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(7/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)
) - (4*(c*e*f + 7*c*d*g - 4*b*e*g)*Sqrt[d + e*x])/(3*c^3*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(
c*e*f + 7*c*d*g - 4*b*e*g)*(d + e*x)^(3/2))/(3*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]
)

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Rubi [A]  time = 0.286686, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {788, 656, 648} \[ \frac{2 (d+e x)^{3/2} (-4 b e g+7 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{4 \sqrt{d+e x} (-4 b e g+7 c d g+c e f)}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)^(7/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(7/2))/(3*c*e^2*(2*c*d - b*e)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2)
) - (4*(c*e*f + 7*c*d*g - 4*b*e*g)*Sqrt[d + e*x])/(3*c^3*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(
c*e*f + 7*c*d*g - 4*b*e*g)*(d + e*x)^(3/2))/(3*c^2*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]
)

Rule 788

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((g*(c*d - b*e) + c*e*f)*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)*(2*c*d - b*e)), x] - Dist[(e*(m*(g
*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c*f - b*g)))/(c*(p + 1)*(2*c*d - b*e)), Int[(d + e*x)^(m - 1)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2,
0] && LtQ[p, -1] && GtQ[m, 0]

Rule 656

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rule 648

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(p + 1)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps

\begin{align*} \int \frac{(d+e x)^{7/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^{7/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{(c e f+7 c d g-4 b e g) \int \frac{(d+e x)^{5/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{7/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}+\frac{2 (c e f+7 c d g-4 b e g) (d+e x)^{3/2}}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(2 (c e f+7 c d g-4 b e g)) \int \frac{(d+e x)^{3/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{3 c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{7/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac{4 (c e f+7 c d g-4 b e g) \sqrt{d+e x}}{3 c^3 e^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (c e f+7 c d g-4 b e g) (d+e x)^{3/2}}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.109789, size = 117, normalized size = 0.54 \[ \frac{2 \sqrt{d+e x} \left (8 b^2 e^2 g-2 b c e (9 d g+e (f-6 g x))+c^2 \left (10 d^2 g+d e (f-15 g x)+3 e^2 x (g x-f)\right )\right )}{3 c^3 e^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)^(7/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[d + e*x]*(8*b^2*e^2*g - 2*b*c*e*(9*d*g + e*(f - 6*g*x)) + c^2*(10*d^2*g + d*e*(f - 15*g*x) + 3*e^2*x*(
-f + g*x))))/(3*c^3*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.006, size = 138, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3\,g{x}^{2}{c}^{2}{e}^{2}+12\,bc{e}^{2}gx-15\,{c}^{2}degx-3\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-18\,bcdeg-2\,bc{e}^{2}f+10\,{c}^{2}{d}^{2}g+{c}^{2}def \right ) }{3\,{c}^{3}{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

2/3*(c*e*x+b*e-c*d)*(3*c^2*e^2*g*x^2+12*b*c*e^2*g*x-15*c^2*d*e*g*x-3*c^2*e^2*f*x+8*b^2*e^2*g-18*b*c*d*e*g-2*b*
c*e^2*f+10*c^2*d^2*g+c^2*d*e*f)*(e*x+d)^(5/2)/c^3/e^2/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)

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Maxima [A]  time = 1.37645, size = 212, normalized size = 0.98 \begin{align*} -\frac{2 \,{\left (3 \, c e x - c d + 2 \, b e\right )} f}{3 \,{\left (c^{3} e^{2} x - c^{3} d e + b c^{2} e^{2}\right )} \sqrt{-c e x + c d - b e}} + \frac{2 \,{\left (3 \, c^{2} e^{2} x^{2} + 10 \, c^{2} d^{2} - 18 \, b c d e + 8 \, b^{2} e^{2} - 3 \,{\left (5 \, c^{2} d e - 4 \, b c e^{2}\right )} x\right )} g}{3 \,{\left (c^{4} e^{3} x - c^{4} d e^{2} + b c^{3} e^{3}\right )} \sqrt{-c e x + c d - b e}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

-2/3*(3*c*e*x - c*d + 2*b*e)*f/((c^3*e^2*x - c^3*d*e + b*c^2*e^2)*sqrt(-c*e*x + c*d - b*e)) + 2/3*(3*c^2*e^2*x
^2 + 10*c^2*d^2 - 18*b*c*d*e + 8*b^2*e^2 - 3*(5*c^2*d*e - 4*b*c*e^2)*x)*g/((c^4*e^3*x - c^4*d*e^2 + b*c^3*e^3)
*sqrt(-c*e*x + c*d - b*e))

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Fricas [A]  time = 1.28126, size = 433, normalized size = 2. \begin{align*} -\frac{2 \,{\left (3 \, c^{2} e^{2} g x^{2} +{\left (c^{2} d e - 2 \, b c e^{2}\right )} f + 2 \,{\left (5 \, c^{2} d^{2} - 9 \, b c d e + 4 \, b^{2} e^{2}\right )} g - 3 \,{\left (c^{2} e^{2} f +{\left (5 \, c^{2} d e - 4 \, b c e^{2}\right )} g\right )} x\right )} \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d}}{3 \,{\left (c^{5} e^{5} x^{3} + c^{5} d^{3} e^{2} - 2 \, b c^{4} d^{2} e^{3} + b^{2} c^{3} d e^{4} -{\left (c^{5} d e^{4} - 2 \, b c^{4} e^{5}\right )} x^{2} -{\left (c^{5} d^{2} e^{3} - b^{2} c^{3} e^{5}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-2/3*(3*c^2*e^2*g*x^2 + (c^2*d*e - 2*b*c*e^2)*f + 2*(5*c^2*d^2 - 9*b*c*d*e + 4*b^2*e^2)*g - 3*(c^2*e^2*f + (5*
c^2*d*e - 4*b*c*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d)/(c^5*e^5*x^3 + c^5*d^3*e^2
- 2*b*c^4*d^2*e^3 + b^2*c^3*d*e^4 - (c^5*d*e^4 - 2*b*c^4*e^5)*x^2 - (c^5*d^2*e^3 - b^2*c^3*e^5)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x