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

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

[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)*Sq
rt[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 = 0.958382, 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 \[ \frac{16 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^4 e^2 \sqrt{d+e x}}+\frac{8 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{15 c^3 e^2}+\frac{2 (d+e x)^{3/2} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+7 c d g+5 c e f)}{5 c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^{7/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]

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)*Sq
rt[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 in Sympy [A]  time = 106.616, size = 280, normalized size = 0.96 \[ \frac{2 \left (d + e x\right )^{\frac{7}{2}} \left (b e g - c d g - c e f\right )}{c e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (6 b e g - 7 c d g - 5 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{5 c^{2} e^{2} \left (b e - 2 c d\right )} - \frac{8 \sqrt{d + e x} \left (6 b e g - 7 c d g - 5 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{15 c^{3} e^{2}} + \frac{16 \left (b e - 2 c d\right ) \left (6 b e g - 7 c d g - 5 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{15 c^{4} e^{2} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

2*(d + e*x)**(7/2)*(b*e*g - c*d*g - c*e*f)/(c*e**2*(b*e - 2*c*d)*sqrt(-b*e**2*x
- c*e**2*x**2 + d*(-b*e + c*d))) + 2*(d + e*x)**(3/2)*(6*b*e*g - 7*c*d*g - 5*c*e
*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(5*c**2*e**2*(b*e - 2*c*d)) -
8*sqrt(d + e*x)*(6*b*e*g - 7*c*d*g - 5*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*
(-b*e + c*d))/(15*c**3*e**2) + 16*(b*e - 2*c*d)*(6*b*e*g - 7*c*d*g - 5*c*e*f)*sq
rt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(15*c**4*e**2*sqrt(d + e*x))

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Mathematica [A]  time = 0.411363, size = 168, normalized size = 0.58 \[ \frac{2 \sqrt{d+e x} \left (-48 b^3 e^3 g+8 b^2 c e^2 (28 d g+5 e f-3 e g x)+2 b c^2 e \left (-167 d^2 g+d e (44 g x-70 f)+e^2 x (10 f+3 g x)\right )+c^3 \left (158 d^3 g+d^2 e (115 f-79 g x)-2 d e^2 x (25 f+8 g x)-e^3 x^2 (5 f+3 g x)\right )\right )}{15 c^4 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]

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

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Maple [A]  time = 0.012, size = 235, normalized size = 0.8 \[{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3\,{e}^{3}g{x}^{3}{c}^{3}-6\,b{c}^{2}{e}^{3}g{x}^{2}+16\,{c}^{3}d{e}^{2}g{x}^{2}+5\,{c}^{3}{e}^{3}f{x}^{2}+24\,{b}^{2}c{e}^{3}gx-88\,b{c}^{2}d{e}^{2}gx-20\,b{c}^{2}{e}^{3}fx+79\,{c}^{3}{d}^{2}egx+50\,{c}^{3}d{e}^{2}fx+48\,{b}^{3}{e}^{3}g-224\,{b}^{2}cd{e}^{2}g-40\,{b}^{2}c{e}^{3}f+334\,b{c}^{2}{d}^{2}eg+140\,b{c}^{2}d{e}^{2}f-158\,{c}^{3}{d}^{3}g-115\,{c}^{3}{d}^{2}ef \right ) }{15\,{c}^{4}{e}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

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 = 0.735209, size = 274, normalized size = 0.94 \[ -\frac{2 \,{\left (c^{2} e^{2} x^{2} - 23 \, c^{2} d^{2} + 28 \, b c d e - 8 \, b^{2} e^{2} + 2 \,{\left (5 \, c^{2} d e - 2 \, b c e^{2}\right )} x\right )} f}{3 \, \sqrt{-c e x + c d - b e} c^{3} e} - \frac{2 \,{\left (3 \, c^{3} e^{3} x^{3} - 158 \, c^{3} d^{3} + 334 \, b c^{2} d^{2} e - 224 \, b^{2} c d e^{2} + 48 \, b^{3} e^{3} + 2 \,{\left (8 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} x^{2} +{\left (79 \, c^{3} d^{2} e - 88 \, b c^{2} d e^{2} + 24 \, b^{2} c e^{3}\right )} x\right )} g}{15 \, \sqrt{-c e x + c d - b e} c^{4} e^{2}} \]

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 + c*d^2 - b*d*e)^(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 = 0.277532, size = 419, normalized size = 1.43 \[ -\frac{2 \,{\left (3 \, c^{3} e^{4} g x^{4} +{\left (5 \, c^{3} e^{4} f +{\left (19 \, c^{3} d e^{3} - 6 \, b c^{2} e^{4}\right )} g\right )} x^{3} +{\left (5 \,{\left (11 \, c^{3} d e^{3} - 4 \, b c^{2} e^{4}\right )} f +{\left (95 \, c^{3} d^{2} e^{2} - 94 \, b c^{2} d e^{3} + 24 \, b^{2} c e^{4}\right )} g\right )} x^{2} - 5 \,{\left (23 \, c^{3} d^{3} e - 28 \, b c^{2} d^{2} e^{2} + 8 \, b^{2} c d e^{3}\right )} f - 2 \,{\left (79 \, c^{3} d^{4} - 167 \, b c^{2} d^{3} e + 112 \, b^{2} c d^{2} e^{2} - 24 \, b^{3} d e^{3}\right )} g -{\left (5 \,{\left (13 \, c^{3} d^{2} e^{2} - 24 \, b c^{2} d e^{3} + 8 \, b^{2} c e^{4}\right )} f +{\left (79 \, c^{3} d^{3} e - 246 \, b c^{2} d^{2} e^{2} + 200 \, b^{2} c d e^{3} - 48 \, b^{3} e^{4}\right )} g\right )} x\right )}}{15 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{4} e^{2}} \]

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 + c*d^2 - b*d*e)^(3/2),x, algorithm="fricas")

[Out]

-2/15*(3*c^3*e^4*g*x^4 + (5*c^3*e^4*f + (19*c^3*d*e^3 - 6*b*c^2*e^4)*g)*x^3 + (5
*(11*c^3*d*e^3 - 4*b*c^2*e^4)*f + (95*c^3*d^2*e^2 - 94*b*c^2*d*e^3 + 24*b^2*c*e^
4)*g)*x^2 - 5*(23*c^3*d^3*e - 28*b*c^2*d^2*e^2 + 8*b^2*c*d*e^3)*f - 2*(79*c^3*d^
4 - 167*b*c^2*d^3*e + 112*b^2*c*d^2*e^2 - 24*b^3*d*e^3)*g - (5*(13*c^3*d^2*e^2 -
24*b*c^2*d*e^3 + 8*b^2*c*e^4)*f + (79*c^3*d^3*e - 246*b*c^2*d^2*e^2 + 200*b^2*c
*d*e^3 - 48*b^3*e^4)*g)*x)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x
+ d)*c^4*e^2)

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

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/XCAS [A]  time = 0.670255, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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 + c*d^2 - b*d*e)^(3/2),x, algorithm="giac")

[Out]

sage0*x