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

Optimal. Leaf size=270 \[ -\frac {16 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^4 e^2 \sqrt {d+e x}}-\frac {8 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^3 e^2}-\frac {2 (d+e x)^{3/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{35 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2} \]

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Rubi [A]  time = 0.46, antiderivative size = 270, normalized size of antiderivative = 1.00, 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 = {794, 656, 648} \begin {gather*} -\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+5 c d g+7 c e f)}{35 c^2 e^2}-\frac {8 \sqrt {d+e x} (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^3 e^2}-\frac {16 (2 c d-b e)^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (-6 b e g+5 c d g+7 c e f)}{105 c^4 e^2 \sqrt {d+e x}}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*(2*c*d - b*e)^2*(7*c*e*f + 5*c*d*g - 6*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*c^4*e^2*Sqr
t[d + e*x]) - (8*(2*c*d - b*e)*(7*c*e*f + 5*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])/(105*c^3*e^2) - (2*(7*c*e*f + 5*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])/(35*c^2*e^2) - (2*g*(d + e*x)^(5/2)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*c*e^2)

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]

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 794

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^{5/2} (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx &=-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}-\frac {\left (2 \left (\frac {1}{2} e \left (-2 c e^2 f+b e^2 g\right )+\frac {5}{2} \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right )\right ) \int \frac {(d+e x)^{5/2}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{7 c e^3}\\ &=-\frac {2 (7 c e f+5 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}}{35 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}+\frac {(4 (2 c d-b e) (7 c e f+5 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}{35 c^2 e}\\ &=-\frac {8 (2 c d-b e) (7 c e f+5 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}}{105 c^3 e^2}-\frac {2 (7 c e f+5 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}}{35 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}+\frac {\left (8 (2 c d-b e)^2 (7 c e f+5 c d g-6 b e g)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{105 c^3 e}\\ &=-\frac {16 (2 c d-b e)^2 (7 c e f+5 c d g-6 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{105 c^4 e^2 \sqrt {d+e x}}-\frac {8 (2 c d-b e) (7 c e f+5 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}}{105 c^3 e^2}-\frac {2 (7 c e f+5 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}}{35 c^2 e^2}-\frac {2 g (d+e x)^{5/2} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{7 c e^2}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 181, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {d+e x} (b e-c d+c e x) \left (-48 b^3 e^3 g+8 b^2 c e^2 (32 d g+7 e f+3 e g x)-2 b c^2 e \left (219 d^2 g+2 d e (63 f+26 g x)+e^2 x (14 f+9 g x)\right )+c^3 \left (230 d^3 g+d^2 e (301 f+115 g x)+2 d e^2 x (49 f+30 g x)+3 e^3 x^2 (7 f+5 g x)\right )\right )}{105 c^4 e^2 \sqrt {(d+e x) (c (d-e x)-b e)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(-(c*d) + b*e + c*e*x)*(-48*b^3*e^3*g + 8*b^2*c*e^2*(7*e*f + 32*d*g + 3*e*g*x) - 2*b*c^2*e*(2
19*d^2*g + e^2*x*(14*f + 9*g*x) + 2*d*e*(63*f + 26*g*x)) + c^3*(230*d^3*g + 3*e^3*x^2*(7*f + 5*g*x) + 2*d*e^2*
x*(49*f + 30*g*x) + d^2*e*(301*f + 115*g*x))))/(105*c^4*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))])

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IntegrateAlgebraic [A]  time = 0.31, size = 248, normalized size = 0.92 \begin {gather*} -\frac {2 \sqrt {(d+e x) (2 c d-b e)-c (d+e x)^2} \left (-48 b^3 e^3 g+24 b^2 c e^2 g (d+e x)+232 b^2 c d e^2 g+56 b^2 c e^3 f-352 b c^2 d^2 e g-28 b c^2 e^2 f (d+e x)-224 b c^2 d e^2 f-18 b c^2 e g (d+e x)^2-68 b c^2 d e g (d+e x)+160 c^3 d^3 g+224 c^3 d^2 e f+40 c^3 d^2 g (d+e x)+21 c^3 e f (d+e x)^2+56 c^3 d e f (d+e x)+15 c^3 g (d+e x)^3+15 c^3 d g (d+e x)^2\right )}{105 c^4 e^2 \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(2*c*d - b*e)*(d + e*x) - c*(d + e*x)^2]*(224*c^3*d^2*e*f - 224*b*c^2*d*e^2*f + 56*b^2*c*e^3*f + 160*
c^3*d^3*g - 352*b*c^2*d^2*e*g + 232*b^2*c*d*e^2*g - 48*b^3*e^3*g + 56*c^3*d*e*f*(d + e*x) - 28*b*c^2*e^2*f*(d
+ e*x) + 40*c^3*d^2*g*(d + e*x) - 68*b*c^2*d*e*g*(d + e*x) + 24*b^2*c*e^2*g*(d + e*x) + 21*c^3*e*f*(d + e*x)^2
+ 15*c^3*d*g*(d + e*x)^2 - 18*b*c^2*e*g*(d + e*x)^2 + 15*c^3*g*(d + e*x)^3))/(105*c^4*e^2*Sqrt[d + e*x])

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fricas [A]  time = 0.40, size = 235, normalized size = 0.87 \begin {gather*} -\frac {2 \, {\left (15 \, c^{3} e^{3} g x^{3} + 3 \, {\left (7 \, c^{3} e^{3} f + 2 \, {\left (10 \, c^{3} d e^{2} - 3 \, b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (43 \, c^{3} d^{2} e - 36 \, b c^{2} d e^{2} + 8 \, b^{2} c e^{3}\right )} f + 2 \, {\left (115 \, c^{3} d^{3} - 219 \, b c^{2} d^{2} e + 128 \, b^{2} c d e^{2} - 24 \, b^{3} e^{3}\right )} g + {\left (14 \, {\left (7 \, c^{3} d e^{2} - 2 \, b c^{2} e^{3}\right )} f + {\left (115 \, c^{3} d^{2} e - 104 \, 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}}{105 \, {\left (c^{4} e^{3} x + c^{4} d e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="fricas")

[Out]

-2/105*(15*c^3*e^3*g*x^3 + 3*(7*c^3*e^3*f + 2*(10*c^3*d*e^2 - 3*b*c^2*e^3)*g)*x^2 + 7*(43*c^3*d^2*e - 36*b*c^2
*d*e^2 + 8*b^2*c*e^3)*f + 2*(115*c^3*d^3 - 219*b*c^2*d^2*e + 128*b^2*c*d*e^2 - 24*b^3*e^3)*g + (14*(7*c^3*d*e^
2 - 2*b*c^2*e^3)*f + (115*c^3*d^2*e - 104*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^4*e^3*x + c^4*d*e^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.05, size = 235, normalized size = 0.87 \begin {gather*} -\frac {2 \left (c e x +b e -c d \right ) \left (-15 g \,e^{3} x^{3} c^{3}+18 b \,c^{2} e^{3} g \,x^{2}-60 c^{3} d \,e^{2} g \,x^{2}-21 c^{3} e^{3} f \,x^{2}-24 b^{2} c \,e^{3} g x +104 b \,c^{2} d \,e^{2} g x +28 b \,c^{2} e^{3} f x -115 c^{3} d^{2} e g x -98 c^{3} d \,e^{2} f x +48 b^{3} e^{3} g -256 b^{2} c d \,e^{2} g -56 b^{2} c \,e^{3} f +438 b \,c^{2} d^{2} e g +252 b \,c^{2} d \,e^{2} f -230 c^{3} d^{3} g -301 f \,d^{2} c^{3} e \right ) \sqrt {e x +d}}{105 \sqrt {-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}}\, c^{4} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-2/105*(c*e*x+b*e-c*d)*(-15*c^3*e^3*g*x^3+18*b*c^2*e^3*g*x^2-60*c^3*d*e^2*g*x^2-21*c^3*e^3*f*x^2-24*b^2*c*e^3*
g*x+104*b*c^2*d*e^2*g*x+28*b*c^2*e^3*f*x-115*c^3*d^2*e*g*x-98*c^3*d*e^2*f*x+48*b^3*e^3*g-256*b^2*c*d*e^2*g-56*
b^2*c*e^3*f+438*b*c^2*d^2*e*g+252*b*c^2*d*e^2*f-230*c^3*d^3*g-301*c^3*d^2*e*f)*(e*x+d)^(1/2)/c^4/e^2/(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(1/2)

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maxima [A]  time = 0.72, size = 319, normalized size = 1.18 \begin {gather*} \frac {2 \, {\left (3 \, c^{3} e^{3} x^{3} - 43 \, c^{3} d^{3} + 79 \, b c^{2} d^{2} e - 44 \, b^{2} c d e^{2} + 8 \, b^{3} e^{3} + {\left (11 \, c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + {\left (29 \, c^{3} d^{2} e - 18 \, b c^{2} d e^{2} + 4 \, b^{2} c e^{3}\right )} x\right )} f}{15 \, \sqrt {-c e x + c d - b e} c^{3} e} + \frac {2 \, {\left (15 \, c^{4} e^{4} x^{4} - 230 \, c^{4} d^{4} + 668 \, b c^{3} d^{3} e - 694 \, b^{2} c^{2} d^{2} e^{2} + 304 \, b^{3} c d e^{3} - 48 \, b^{4} e^{4} + 3 \, {\left (15 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + {\left (55 \, c^{4} d^{2} e^{2} - 26 \, b c^{3} d e^{3} + 6 \, b^{2} c^{2} e^{4}\right )} x^{2} + {\left (115 \, c^{4} d^{3} e - 219 \, b c^{3} d^{2} e^{2} + 128 \, b^{2} c^{2} d e^{3} - 24 \, b^{3} c e^{4}\right )} x\right )} g}{105 \, \sqrt {-c e x + c d - b e} c^{4} e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algorithm="maxima")

[Out]

2/15*(3*c^3*e^3*x^3 - 43*c^3*d^3 + 79*b*c^2*d^2*e - 44*b^2*c*d*e^2 + 8*b^3*e^3 + (11*c^3*d*e^2 - b*c^2*e^3)*x^
2 + (29*c^3*d^2*e - 18*b*c^2*d*e^2 + 4*b^2*c*e^3)*x)*f/(sqrt(-c*e*x + c*d - b*e)*c^3*e) + 2/105*(15*c^4*e^4*x^
4 - 230*c^4*d^4 + 668*b*c^3*d^3*e - 694*b^2*c^2*d^2*e^2 + 304*b^3*c*d*e^3 - 48*b^4*e^4 + 3*(15*c^4*d*e^3 - b*c
^3*e^4)*x^3 + (55*c^4*d^2*e^2 - 26*b*c^3*d*e^3 + 6*b^2*c^2*e^4)*x^2 + (115*c^4*d^3*e - 219*b*c^3*d^2*e^2 + 128
*b^2*c^2*d*e^3 - 24*b^3*c*e^4)*x)*g/(sqrt(-c*e*x + c*d - b*e)*c^4*e^2)

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mupad [B]  time = 2.77, size = 246, normalized size = 0.91 \begin {gather*} -\frac {\left (\frac {2\,g\,x^3\,\sqrt {d+e\,x}}{7\,c}+\frac {\sqrt {d+e\,x}\,\left (-96\,g\,b^3\,e^3+512\,g\,b^2\,c\,d\,e^2+112\,f\,b^2\,c\,e^3-876\,g\,b\,c^2\,d^2\,e-504\,f\,b\,c^2\,d\,e^2+460\,g\,c^3\,d^3+602\,f\,c^3\,d^2\,e\right )}{105\,c^4\,e^3}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (20\,c\,d\,g-6\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^2\,e}+\frac {x\,\sqrt {d+e\,x}\,\left (48\,g\,b^2\,c\,e^3-208\,g\,b\,c^2\,d\,e^2-56\,f\,b\,c^2\,e^3+230\,g\,c^3\,d^2\,e+196\,f\,c^3\,d\,e^2\right )}{105\,c^4\,e^3}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{x+\frac {d}{e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(d + e*x)^(5/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2),x)

[Out]

-(((2*g*x^3*(d + e*x)^(1/2))/(7*c) + ((d + e*x)^(1/2)*(460*c^3*d^3*g - 96*b^3*e^3*g + 112*b^2*c*e^3*f + 602*c^
3*d^2*e*f - 504*b*c^2*d*e^2*f - 876*b*c^2*d^2*e*g + 512*b^2*c*d*e^2*g))/(105*c^4*e^3) + (2*x^2*(d + e*x)^(1/2)
*(20*c*d*g - 6*b*e*g + 7*c*e*f))/(35*c^2*e) + (x*(d + e*x)^(1/2)*(48*b^2*c*e^3*g - 56*b*c^2*e^3*f + 196*c^3*d*
e^2*f + 230*c^3*d^2*e*g - 208*b*c^2*d*e^2*g))/(105*c^4*e^3))*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(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 {5}{2}} \left (f + g x\right )}{\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(5/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

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

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