3.2269 \(\int \frac{(d+e x)^{7/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=292 \[ \frac{2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{5 c^2 e^2 (2 c d-b e)}+\frac{8 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^3 e^2}+\frac{16 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^4 e^2 \sqrt{d+e x}}+\frac{2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(7/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) +
(16*(2*c*d - b*e)*(5*c*e*f + 7*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*c^4*e^2*Sqrt[d
+ e*x]) + (8*(5*c*e*f + 7*c*d*g - 6*b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*c^3*e^
2) + (2*(5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(5*c^2*e^2*(2
*c*d - b*e))

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Rubi [A]  time = 0.414253, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {788, 656, 648} \[ \frac{2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{5 c^2 e^2 (2 c d-b e)}+\frac{8 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^3 e^2}+\frac{16 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^4 e^2 \sqrt{d+e x}}+\frac{2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

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)^(3/2),x]

[Out]

(2*(c*e*f + c*d*g - b*e*g)*(d + e*x)^(7/2))/(c*e^2*(2*c*d - b*e)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) +
(16*(2*c*d - b*e)*(5*c*e*f + 7*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*c^4*e^2*Sqrt[d
+ e*x]) + (8*(5*c*e*f + 7*c*d*g - 6*b*e*g)*Sqrt[d + e*x]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(15*c^3*e^
2) + (2*(5*c*e*f + 7*c*d*g - 6*b*e*g)*(d + e*x)^(3/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(5*c^2*e^2*(2
*c*d - b*e))

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 )^{3/2}} \, dx &=\frac{2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}-\frac{(5 c e f+7 c d g-6 b e g) \int \frac{(d+e x)^{5/2}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{c e (2 c d-b e)}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}-\frac{(4 (5 c e f+7 c d g-6 b e g)) \int \frac{(d+e x)^{3/2}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{5 c^2 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{8 (5 c e f+7 c d g-6 b e g) \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^3 e^2}+\frac{2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}-\frac{(8 (2 c d-b e) (5 c e f+7 c d g-6 b e g)) \int \frac{\sqrt{d+e x}}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{15 c^3 e}\\ &=\frac{2 (c e f+c d g-b e g) (d+e x)^{7/2}}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{16 (2 c d-b e) (5 c e f+7 c d g-6 b e g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^4 e^2 \sqrt{d+e x}}+\frac{8 (5 c e f+7 c d g-6 b e g) \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{15 c^3 e^2}+\frac{2 (5 c e f+7 c d g-6 b e g) (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{5 c^2 e^2 (2 c d-b e)}\\ \end{align*}

Mathematica [A]  time = 0.146365, size = 168, normalized size = 0.58 \[ -\frac{2 \sqrt{d+e x} \left (-8 b^2 c e^2 (28 d g+5 e f-3 e g x)+48 b^3 e^3 g+2 b c^2 e \left (167 d^2 g+d e (70 f-44 g x)-e^2 x (10 f+3 g x)\right )+c^3 \left (d^2 e (79 g x-115 f)-158 d^3 g+2 d e^2 x (25 f+8 g x)+e^3 x^2 (5 f+3 g x)\right )\right )}{15 c^4 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

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)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(48*b^3*e^3*g - 8*b^2*c*e^2*(5*e*f + 28*d*g - 3*e*g*x) + 2*b*c^2*e*(167*d^2*g + d*e*(70*f -
44*g*x) - e^2*x*(10*f + 3*g*x)) + c^3*(-158*d^3*g + e^3*x^2*(5*f + 3*g*x) + 2*d*e^2*x*(25*f + 8*g*x) + d^2*e*(
-115*f + 79*g*x))))/(15*c^4*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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Maple [A]  time = 0.007, size = 235, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3\,g{e}^{3}{x}^{3}{c}^{3}-6\,b{c}^{2}{e}^{3}g{x}^{2}+16\,{c}^{3}d{e}^{2}g{x}^{2}+5\,{c}^{3}{e}^{3}f{x}^{2}+24\,{b}^{2}c{e}^{3}gx-88\,b{c}^{2}d{e}^{2}gx-20\,b{c}^{2}{e}^{3}fx+79\,{c}^{3}{d}^{2}egx+50\,{c}^{3}d{e}^{2}fx+48\,{b}^{3}{e}^{3}g-224\,{b}^{2}cd{e}^{2}g-40\,{b}^{2}c{e}^{3}f+334\,b{c}^{2}{d}^{2}eg+140\,b{c}^{2}d{e}^{2}f-158\,{c}^{3}{d}^{3}g-115\,{c}^{3}{d}^{2}ef \right ) }{15\,{c}^{4}{e}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{3}{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)^(3/2),x)

[Out]

2/15*(c*e*x+b*e-c*d)*(3*c^3*e^3*g*x^3-6*b*c^2*e^3*g*x^2+16*c^3*d*e^2*g*x^2+5*c^3*e^3*f*x^2+24*b^2*c*e^3*g*x-88
*b*c^2*d*e^2*g*x-20*b*c^2*e^3*f*x+79*c^3*d^2*e*g*x+50*c^3*d*e^2*f*x+48*b^3*e^3*g-224*b^2*c*d*e^2*g-40*b^2*c*e^
3*f+334*b*c^2*d^2*e*g+140*b*c^2*d*e^2*f-158*c^3*d^3*g-115*c^3*d^2*e*f)*(e*x+d)^(3/2)/c^4/e^2/(-c*e^2*x^2-b*e^2
*x-b*d*e+c*d^2)^(3/2)

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Maxima [A]  time = 1.19114, size = 274, normalized size = 0.94 \begin{align*} -\frac{2 \,{\left (c^{2} e^{2} x^{2} - 23 \, c^{2} d^{2} + 28 \, b c d e - 8 \, b^{2} e^{2} + 2 \,{\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \, \sqrt{-c e x + c d - b e} c^{3} e} - \frac{2 \,{\left (3 \, c^{3} e^{3} x^{3} - 158 \, c^{3} d^{3} + 334 \, b c^{2} d^{2} e - 224 \, b^{2} c d e^{2} + 48 \, b^{3} e^{3} + 2 \,{\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} x^{2} +{\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} x\right )} g}{15 \, \sqrt{-c e x + c d - b e} c^{4} e^{2}} \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)^(3/2),x, algorithm="maxima")

[Out]

-2/3*(c^2*e^2*x^2 - 23*c^2*d^2 + 28*b*c*d*e - 8*b^2*e^2 + 2*(5*c^2*d*e - 2*b*c*e^2)*x)*f/(sqrt(-c*e*x + c*d -
b*e)*c^3*e) - 2/15*(3*c^3*e^3*x^3 - 158*c^3*d^3 + 334*b*c^2*d^2*e - 224*b^2*c*d*e^2 + 48*b^3*e^3 + 2*(8*c^3*d*
e^2 - 3*b*c^2*e^3)*x^2 + (79*c^3*d^2*e - 88*b*c^2*d*e^2 + 24*b^2*c*e^3)*x)*g/(sqrt(-c*e*x + c*d - b*e)*c^4*e^2
)

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Fricas [A]  time = 1.38814, size = 541, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (3 \, c^{3} e^{3} g x^{3} +{\left (5 \, c^{3} e^{3} f + 2 \,{\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} g\right )} x^{2} - 5 \,{\left (23 \, c^{3} d^{2} e - 28 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f - 2 \,{\left (79 \, c^{3} d^{3} - 167 \, b c^{2} d^{2} e + 112 \, b^{2} c d e^{2} - 24 \, b^{3} e^{3}\right )} g +{\left (10 \,{\left (5 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f +{\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\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}}{15 \,{\left (c^{5} e^{4} x^{2} + b c^{4} e^{4} x - c^{5} d^{2} e^{2} + b c^{4} d e^{3}\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)^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*c^3*e^3*g*x^3 + (5*c^3*e^3*f + 2*(8*c^3*d*e^2 - 3*b*c^2*e^3)*g)*x^2 - 5*(23*c^3*d^2*e - 28*b*c^2*d*e^2
+ 8*b^2*c*e^3)*f - 2*(79*c^3*d^3 - 167*b*c^2*d^2*e + 112*b^2*c*d*e^2 - 24*b^3*e^3)*g + (10*(5*c^3*d*e^2 - 2*b
*c^2*e^3)*f + (79*c^3*d^2*e - 88*b*c^2*d*e^2 + 24*b^2*c*e^3)*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^4*x^2 + b*c^4*e^4*x - c^5*d^2*e^2 + b*c^4*d*e^3)

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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)**(3/2),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)^(3/2),x, algorithm="giac")

[Out]

sage0*x