\(\int \frac {(d+e x)^{13/2} (f+g x)}{(c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2}} \, dx\) [2276]

Optimal result

Integrand size = 46, antiderivative size = 448 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}-\frac {256 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g) \sqrt {d+e x}}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}} \]

[Out]

2/3*(-b*e*g+c*d*g+c*e*f)*(e*x+d)^(13/2)/c/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)+128/105*(-b*
e+2*c*d)^2*(-10*b*e*g+13*c*d*g+7*c*e*f)*(e*x+d)^(3/2)/c^5/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+32/105*(-
b*e+2*c*d)*(-10*b*e*g+13*c*d*g+7*c*e*f)*(e*x+d)^(5/2)/c^4/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+16/105*(-
10*b*e*g+13*c*d*g+7*c*e*f)*(e*x+d)^(7/2)/c^3/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)+2/21*(-10*b*e*g+13*c*d
*g+7*c*e*f)*(e*x+d)^(9/2)/c^2/e^2/(-b*e+2*c*d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)-256/105*(-b*e+2*c*d)^3*(
-10*b*e*g+13*c*d*g+7*c*e*f)*(e*x+d)^(1/2)/c^6/e^2/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {802, 670, 662} \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {256 \sqrt {d+e x} (2 c d-b e)^3 (-10 b e g+13 c d g+7 c e f)}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (d+e x)^{3/2} (2 c d-b e)^2 (-10 b e g+13 c d g+7 c e f)}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (d+e x)^{5/2} (2 c d-b e) (-10 b e g+13 c d g+7 c e f)}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (d+e x)^{7/2} (-10 b e g+13 c d g+7 c e f)}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{9/2} (-10 b e g+13 c d g+7 c e f)}{21 c^2 e^2 (2 c d-b e) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (d+e x)^{13/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]

[In]

Int[((d + e*x)^(13/2)*(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)^(13/2))/(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
)) - (256*(2*c*d - b*e)^3*(7*c*e*f + 13*c*d*g - 10*b*e*g)*Sqrt[d + e*x])/(105*c^6*e^2*Sqrt[d*(c*d - b*e) - b*e
^2*x - c*e^2*x^2]) + (128*(2*c*d - b*e)^2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(3/2))/(105*c^5*e^2*Sqrt[d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (32*(2*c*d - b*e)*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(5/2))/(105
*c^4*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (16*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(7/2))/(10
5*c^3*e^2*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2]) + (2*(7*c*e*f + 13*c*d*g - 10*b*e*g)*(d + e*x)^(9/2))/(21
*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 662

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 670

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 802

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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 {(7 c e f+13 c d g-10 b e g) \int \frac {(d+e x)^{11/2}}{\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)^{13/2}}{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 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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 {(8 (7 c e f+13 c d g-10 b e g)) \int \frac {(d+e x)^{9/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{21 c^2 e} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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 {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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 {(16 (2 c d-b e) (7 c e f+13 c d g-10 b e g)) \int \frac {(d+e x)^{7/2}}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}} \, dx}{35 c^3 e} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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 {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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 {\left (64 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g)\right ) \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}{105 c^4 e} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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 {128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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 {\left (128 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g)\right ) \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}{105 c^5 e} \\ & = \frac {2 (c e f+c d g-b e g) (d+e x)^{13/2}}{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 {256 (2 c d-b e)^3 (7 c e f+13 c d g-10 b e g) \sqrt {d+e x}}{105 c^6 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {128 (2 c d-b e)^2 (7 c e f+13 c d g-10 b e g) (d+e x)^{3/2}}{105 c^5 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {32 (2 c d-b e) (7 c e f+13 c d g-10 b e g) (d+e x)^{5/2}}{105 c^4 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {16 (7 c e f+13 c d g-10 b e g) (d+e x)^{7/2}}{105 c^3 e^2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac {2 (7 c e f+13 c d g-10 b e g) (d+e x)^{9/2}}{21 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] (verified)

Time = 0.28 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.82 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \sqrt {d+e x} \left (-1280 b^5 e^5 g+128 b^4 c e^4 (7 e f+78 d g-15 e g x)-32 b^3 c^2 e^3 \left (953 d^2 g+2 d e (91 f-204 g x)+3 e^2 x (-14 f+5 g x)\right )+16 b^2 c^3 e^2 \left (2844 d^3 g+3 d^2 e (287 f-681 g x)+e^3 x^2 (21 f+5 g x)+6 d e^2 x (-77 f+29 g x)\right )+c^5 \left (9414 d^5 g+3 d^4 e (1687 f-4707 g x)+3 e^5 x^4 (7 f+5 g x)+2 d e^4 x^3 (98 f+57 g x)+2 d^2 e^3 x^2 (903 f+257 g x)+12 d^3 e^2 x (-637 f+292 g x)\right )-2 b c^4 e \left (16563 d^4 g+12 d^3 e (581 f-1482 g x)+e^4 x^3 (28 f+15 g x)+12 d e^3 x^2 (63 f+16 g x)+6 d^2 e^2 x (-1106 f+449 g x)\right )\right )}{105 c^6 e^2 (-c d+b e+c e x) \sqrt {(d+e x) (-b e+c (d-e x))}} \]

[In]

Integrate[((d + e*x)^(13/2)*(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]*(-1280*b^5*e^5*g + 128*b^4*c*e^4*(7*e*f + 78*d*g - 15*e*g*x) - 32*b^3*c^2*e^3*(953*d^2*g + 2*
d*e*(91*f - 204*g*x) + 3*e^2*x*(-14*f + 5*g*x)) + 16*b^2*c^3*e^2*(2844*d^3*g + 3*d^2*e*(287*f - 681*g*x) + e^3
*x^2*(21*f + 5*g*x) + 6*d*e^2*x*(-77*f + 29*g*x)) + c^5*(9414*d^5*g + 3*d^4*e*(1687*f - 4707*g*x) + 3*e^5*x^4*
(7*f + 5*g*x) + 2*d*e^4*x^3*(98*f + 57*g*x) + 2*d^2*e^3*x^2*(903*f + 257*g*x) + 12*d^3*e^2*x*(-637*f + 292*g*x
)) - 2*b*c^4*e*(16563*d^4*g + 12*d^3*e*(581*f - 1482*g*x) + e^4*x^3*(28*f + 15*g*x) + 12*d*e^3*x^2*(63*f + 16*
g*x) + 6*d^2*e^2*x*(-1106*f + 449*g*x))))/(105*c^6*e^2*(-(c*d) + b*e + c*e*x)*Sqrt[(d + e*x)*(-(b*e) + c*(d -
e*x))])

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.18

method	result	size
default	\(\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (-15 g \,e^{5} x^{5} c^{5}+30 b \,c^{4} e^{5} g \,x^{4}-114 c^{5} d \,e^{4} g \,x^{4}-21 c^{5} e^{5} f \,x^{4}-80 b^{2} c^{3} e^{5} g \,x^{3}+384 b \,c^{4} d \,e^{4} g \,x^{3}+56 b \,c^{4} e^{5} f \,x^{3}-514 c^{5} d^{2} e^{3} g \,x^{3}-196 c^{5} d \,e^{4} f \,x^{3}+480 b^{3} c^{2} e^{5} g \,x^{2}-2784 b^{2} c^{3} d \,e^{4} g \,x^{2}-336 b^{2} c^{3} e^{5} f \,x^{2}+5388 b \,c^{4} d^{2} e^{3} g \,x^{2}+1512 b \,c^{4} d \,e^{4} f \,x^{2}-3504 c^{5} d^{3} e^{2} g \,x^{2}-1806 c^{5} d^{2} e^{3} f \,x^{2}+1920 b^{4} c \,e^{5} g x -13056 b^{3} c^{2} d \,e^{4} g x -1344 b^{3} c^{2} e^{5} f x +32688 b^{2} c^{3} d^{2} e^{3} g x +7392 b^{2} c^{3} d \,e^{4} f x -35568 b \,c^{4} d^{3} e^{2} g x -13272 b \,c^{4} d^{2} e^{3} f x +14121 c^{5} d^{4} e g x +7644 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -9984 b^{4} c d \,e^{4} g -896 b^{4} c \,e^{5} f +30496 b^{3} c^{2} d^{2} e^{3} g +5824 b^{3} c^{2} d \,e^{4} f -45504 b^{2} c^{3} d^{3} e^{2} g -13776 b^{2} c^{3} d^{2} e^{3} f +33126 b \,c^{4} d^{4} e g +13944 b \,c^{4} d^{3} e^{2} f -9414 c^{5} d^{5} g -5061 d^{4} f \,c^{5} e \right )}{105 \sqrt {e x +d}\, \left (x c e +b e -c d \right )^{2} c^{6} e^{2}}\)	\(529\)
gosper	\(-\frac {2 \left (x c e +b e -c d \right ) \left (-15 g \,e^{5} x^{5} c^{5}+30 b \,c^{4} e^{5} g \,x^{4}-114 c^{5} d \,e^{4} g \,x^{4}-21 c^{5} e^{5} f \,x^{4}-80 b^{2} c^{3} e^{5} g \,x^{3}+384 b \,c^{4} d \,e^{4} g \,x^{3}+56 b \,c^{4} e^{5} f \,x^{3}-514 c^{5} d^{2} e^{3} g \,x^{3}-196 c^{5} d \,e^{4} f \,x^{3}+480 b^{3} c^{2} e^{5} g \,x^{2}-2784 b^{2} c^{3} d \,e^{4} g \,x^{2}-336 b^{2} c^{3} e^{5} f \,x^{2}+5388 b \,c^{4} d^{2} e^{3} g \,x^{2}+1512 b \,c^{4} d \,e^{4} f \,x^{2}-3504 c^{5} d^{3} e^{2} g \,x^{2}-1806 c^{5} d^{2} e^{3} f \,x^{2}+1920 b^{4} c \,e^{5} g x -13056 b^{3} c^{2} d \,e^{4} g x -1344 b^{3} c^{2} e^{5} f x +32688 b^{2} c^{3} d^{2} e^{3} g x +7392 b^{2} c^{3} d \,e^{4} f x -35568 b \,c^{4} d^{3} e^{2} g x -13272 b \,c^{4} d^{2} e^{3} f x +14121 c^{5} d^{4} e g x +7644 c^{5} d^{3} e^{2} f x +1280 b^{5} e^{5} g -9984 b^{4} c d \,e^{4} g -896 b^{4} c \,e^{5} f +30496 b^{3} c^{2} d^{2} e^{3} g +5824 b^{3} c^{2} d \,e^{4} f -45504 b^{2} c^{3} d^{3} e^{2} g -13776 b^{2} c^{3} d^{2} e^{3} f +33126 b \,c^{4} d^{4} e g +13944 b \,c^{4} d^{3} e^{2} f -9414 c^{5} d^{5} g -5061 d^{4} f \,c^{5} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{105 c^{6} e^{2} \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {5}{2}}}\)	\(535\)

[In]

int((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/105/(e*x+d)^(1/2)*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(-15*c^5*e^5*g*x^5+30*b*c^4*e^5*g*x^4-114*c^5*d*e^4*g*x^4
-21*c^5*e^5*f*x^4-80*b^2*c^3*e^5*g*x^3+384*b*c^4*d*e^4*g*x^3+56*b*c^4*e^5*f*x^3-514*c^5*d^2*e^3*g*x^3-196*c^5*
d*e^4*f*x^3+480*b^3*c^2*e^5*g*x^2-2784*b^2*c^3*d*e^4*g*x^2-336*b^2*c^3*e^5*f*x^2+5388*b*c^4*d^2*e^3*g*x^2+1512
*b*c^4*d*e^4*f*x^2-3504*c^5*d^3*e^2*g*x^2-1806*c^5*d^2*e^3*f*x^2+1920*b^4*c*e^5*g*x-13056*b^3*c^2*d*e^4*g*x-13
44*b^3*c^2*e^5*f*x+32688*b^2*c^3*d^2*e^3*g*x+7392*b^2*c^3*d*e^4*f*x-35568*b*c^4*d^3*e^2*g*x-13272*b*c^4*d^2*e^
3*f*x+14121*c^5*d^4*e*g*x+7644*c^5*d^3*e^2*f*x+1280*b^5*e^5*g-9984*b^4*c*d*e^4*g-896*b^4*c*e^5*f+30496*b^3*c^2
*d^2*e^3*g+5824*b^3*c^2*d*e^4*f-45504*b^2*c^3*d^3*e^2*g-13776*b^2*c^3*d^2*e^3*f+33126*b*c^4*d^4*e*g+13944*b*c^
4*d^3*e^2*f-9414*c^5*d^5*g-5061*c^5*d^4*e*f)/(c*e*x+b*e-c*d)^2/c^6/e^2

Fricas [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (15 \, c^{5} e^{5} g x^{5} + 3 \, {\left (7 \, c^{5} e^{5} f + 2 \, {\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} g\right )} x^{4} + 2 \, {\left (14 \, {\left (7 \, c^{5} d e^{4} - 2 \, b c^{4} e^{5}\right )} f + {\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} g\right )} x^{3} + 6 \, {\left (7 \, {\left (43 \, c^{5} d^{2} e^{3} - 36 \, b c^{4} d e^{4} + 8 \, b^{2} c^{3} e^{5}\right )} f + 2 \, {\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} g\right )} x^{2} + 7 \, {\left (723 \, c^{5} d^{4} e - 1992 \, b c^{4} d^{3} e^{2} + 1968 \, b^{2} c^{3} d^{2} e^{3} - 832 \, b^{3} c^{2} d e^{4} + 128 \, b^{4} c e^{5}\right )} f + 2 \, {\left (4707 \, c^{5} d^{5} - 16563 \, b c^{4} d^{4} e + 22752 \, b^{2} c^{3} d^{3} e^{2} - 15248 \, b^{3} c^{2} d^{2} e^{3} + 4992 \, b^{4} c d e^{4} - 640 \, b^{5} e^{5}\right )} g - 3 \, {\left (28 \, {\left (91 \, c^{5} d^{3} e^{2} - 158 \, b c^{4} d^{2} e^{3} + 88 \, b^{2} c^{3} d e^{4} - 16 \, b^{3} c^{2} e^{5}\right )} f + {\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{105 \, {\left (c^{8} e^{5} x^{3} + c^{8} d^{3} e^{2} - 2 \, b c^{7} d^{2} e^{3} + b^{2} c^{6} d e^{4} - {\left (c^{8} d e^{4} - 2 \, b c^{7} e^{5}\right )} x^{2} - {\left (c^{8} d^{2} e^{3} - b^{2} c^{6} e^{5}\right )} x\right )}} \]

[In]

integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

-2/105*(15*c^5*e^5*g*x^5 + 3*(7*c^5*e^5*f + 2*(19*c^5*d*e^4 - 5*b*c^4*e^5)*g)*x^4 + 2*(14*(7*c^5*d*e^4 - 2*b*c
^4*e^5)*f + (257*c^5*d^2*e^3 - 192*b*c^4*d*e^4 + 40*b^2*c^3*e^5)*g)*x^3 + 6*(7*(43*c^5*d^2*e^3 - 36*b*c^4*d*e^
4 + 8*b^2*c^3*e^5)*f + 2*(292*c^5*d^3*e^2 - 449*b*c^4*d^2*e^3 + 232*b^2*c^3*d*e^4 - 40*b^3*c^2*e^5)*g)*x^2 + 7
*(723*c^5*d^4*e - 1992*b*c^4*d^3*e^2 + 1968*b^2*c^3*d^2*e^3 - 832*b^3*c^2*d*e^4 + 128*b^4*c*e^5)*f + 2*(4707*c
^5*d^5 - 16563*b*c^4*d^4*e + 22752*b^2*c^3*d^3*e^2 - 15248*b^3*c^2*d^2*e^3 + 4992*b^4*c*d*e^4 - 640*b^5*e^5)*g
- 3*(28*(91*c^5*d^3*e^2 - 158*b*c^4*d^2*e^3 + 88*b^2*c^3*d*e^4 - 16*b^3*c^2*e^5)*f + (4707*c^5*d^4*e - 11856*
b*c^4*d^3*e^2 + 10896*b^2*c^3*d^2*e^3 - 4352*b^3*c^2*d*e^4 + 640*b^4*c*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e)*sqrt(e*x + d)/(c^8*e^5*x^3 + c^8*d^3*e^2 - 2*b*c^7*d^2*e^3 + b^2*c^6*d*e^4 - (c^8*d*e^4 - 2*b*c
^7*e^5)*x^2 - (c^8*d^2*e^3 - b^2*c^6*e^5)*x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(13/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.14 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (3 \, c^{4} e^{4} x^{4} + 723 \, c^{4} d^{4} - 1992 \, b c^{3} d^{3} e + 1968 \, b^{2} c^{2} d^{2} e^{2} - 832 \, b^{3} c d e^{3} + 128 \, b^{4} e^{4} + 4 \, {\left (7 \, c^{4} d e^{3} - 2 \, b c^{3} e^{4}\right )} x^{3} + 6 \, {\left (43 \, c^{4} d^{2} e^{2} - 36 \, b c^{3} d e^{3} + 8 \, b^{2} c^{2} e^{4}\right )} x^{2} - 12 \, {\left (91 \, c^{4} d^{3} e - 158 \, b c^{3} d^{2} e^{2} + 88 \, b^{2} c^{2} d e^{3} - 16 \, b^{3} c e^{4}\right )} x\right )} f}{15 \, {\left (c^{6} e^{2} x - c^{6} d e + b c^{5} e^{2}\right )} \sqrt {-c e x + c d - b e}} + \frac {2 \, {\left (15 \, c^{5} e^{5} x^{5} + 9414 \, c^{5} d^{5} - 33126 \, b c^{4} d^{4} e + 45504 \, b^{2} c^{3} d^{3} e^{2} - 30496 \, b^{3} c^{2} d^{2} e^{3} + 9984 \, b^{4} c d e^{4} - 1280 \, b^{5} e^{5} + 6 \, {\left (19 \, c^{5} d e^{4} - 5 \, b c^{4} e^{5}\right )} x^{4} + 2 \, {\left (257 \, c^{5} d^{2} e^{3} - 192 \, b c^{4} d e^{4} + 40 \, b^{2} c^{3} e^{5}\right )} x^{3} + 12 \, {\left (292 \, c^{5} d^{3} e^{2} - 449 \, b c^{4} d^{2} e^{3} + 232 \, b^{2} c^{3} d e^{4} - 40 \, b^{3} c^{2} e^{5}\right )} x^{2} - 3 \, {\left (4707 \, c^{5} d^{4} e - 11856 \, b c^{4} d^{3} e^{2} + 10896 \, b^{2} c^{3} d^{2} e^{3} - 4352 \, b^{3} c^{2} d e^{4} + 640 \, b^{4} c e^{5}\right )} x\right )} g}{105 \, {\left (c^{7} e^{3} x - c^{7} d e^{2} + b c^{6} e^{3}\right )} \sqrt {-c e x + c d - b e}} \]

[In]

integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

2/15*(3*c^4*e^4*x^4 + 723*c^4*d^4 - 1992*b*c^3*d^3*e + 1968*b^2*c^2*d^2*e^2 - 832*b^3*c*d*e^3 + 128*b^4*e^4 +
4*(7*c^4*d*e^3 - 2*b*c^3*e^4)*x^3 + 6*(43*c^4*d^2*e^2 - 36*b*c^3*d*e^3 + 8*b^2*c^2*e^4)*x^2 - 12*(91*c^4*d^3*e
- 158*b*c^3*d^2*e^2 + 88*b^2*c^2*d*e^3 - 16*b^3*c*e^4)*x)*f/((c^6*e^2*x - c^6*d*e + b*c^5*e^2)*sqrt(-c*e*x +
c*d - b*e)) + 2/105*(15*c^5*e^5*x^5 + 9414*c^5*d^5 - 33126*b*c^4*d^4*e + 45504*b^2*c^3*d^3*e^2 - 30496*b^3*c^2
*d^2*e^3 + 9984*b^4*c*d*e^4 - 1280*b^5*e^5 + 6*(19*c^5*d*e^4 - 5*b*c^4*e^5)*x^4 + 2*(257*c^5*d^2*e^3 - 192*b*c
^4*d*e^4 + 40*b^2*c^3*e^5)*x^3 + 12*(292*c^5*d^3*e^2 - 449*b*c^4*d^2*e^3 + 232*b^2*c^3*d*e^4 - 40*b^3*c^2*e^5)
*x^2 - 3*(4707*c^5*d^4*e - 11856*b*c^4*d^3*e^2 + 10896*b^2*c^3*d^2*e^3 - 4352*b^3*c^2*d*e^4 + 640*b^4*c*e^5)*x
)*g/((c^7*e^3*x - c^7*d*e^2 + b*c^6*e^3)*sqrt(-c*e*x + c*d - b*e))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1092 vs. \(2 (412) = 824\).

Time = 0.37 (sec) , antiderivative size = 1092, normalized size of antiderivative = 2.44 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(13/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

256/105*(56*c^4*d^3*e*f - 84*b*c^3*d^2*e^2*f + 42*b^2*c^2*d*e^3*f - 7*b^3*c*e^4*f + 104*c^4*d^4*g - 236*b*c^3*
d^3*e*g + 198*b^2*c^2*d^2*e^2*g - 73*b^3*c*d*e^3*g + 10*b^4*e^4*g)/(sqrt(2*c*d - b*e)*c^6*e^2) - 2/3*(16*c^5*d
^4*e*f - 32*b*c^4*d^3*e^2*f + 24*b^2*c^3*d^2*e^3*f - 8*b^3*c^2*d*e^4*f + b^4*c*e^5*f + 16*c^5*d^5*g - 48*b*c^4
*d^4*e*g + 56*b^2*c^3*d^3*e^2*g - 32*b^3*c^2*d^2*e^3*g + 9*b^4*c*d*e^4*g - b^5*e^5*g + 96*((e*x + d)*c - 2*c*d
+ b*e)*c^4*d^3*e*f - 144*((e*x + d)*c - 2*c*d + b*e)*b*c^3*d^2*e^2*f + 72*((e*x + d)*c - 2*c*d + b*e)*b^2*c^2
*d*e^3*f - 12*((e*x + d)*c - 2*c*d + b*e)*b^3*c*e^4*f + 144*((e*x + d)*c - 2*c*d + b*e)*c^4*d^4*g - 336*((e*x
+ d)*c - 2*c*d + b*e)*b*c^3*d^3*e*g + 288*((e*x + d)*c - 2*c*d + b*e)*b^2*c^2*d^2*e^2*g - 108*((e*x + d)*c - 2
*c*d + b*e)*b^3*c*d*e^3*g + 15*((e*x + d)*c - 2*c*d + b*e)*b^4*e^4*g)/(((e*x + d)*c - 2*c*d + b*e)*sqrt(-(e*x
+ d)*c + 2*c*d - b*e)*c^6*e^2) - 2/105*(2520*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^39*d^2*e^13*f - 2520*sqrt(-(e*
x + d)*c + 2*c*d - b*e)*b*c^38*d*e^14*f + 630*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^37*e^15*f + 5880*sqrt(-(e
*x + d)*c + 2*c*d - b*e)*c^39*d^3*e^12*g - 10080*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^38*d^2*e^13*g + 5670*sqr
t(-(e*x + d)*c + 2*c*d - b*e)*b^2*c^37*d*e^14*g - 1050*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^3*c^36*e^15*g - 280*
(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^38*d*e^13*f + 140*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^37*e^14*f - 1120
*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^38*d^2*e^12*g + 1260*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^37*d*e^13*g
- 350*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*c^36*e^14*g + 21*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c
+ 2*c*d - b*e)*c^37*e^13*f + 189*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^37*d*e^12*g
- 105*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^36*e^13*g + 15*((e*x + d)*c - 2*c*d +
b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^36*e^12*g)/(c^42*e^14)

Mupad [B] (verification not implemented)

Time = 12.41 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.33 \[ \int \frac {(d+e x)^{13/2} (f+g x)}{\left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}\,\left (\frac {\sqrt {d+e\,x}\,\left (-2560\,g\,b^5\,e^5+19968\,g\,b^4\,c\,d\,e^4+1792\,f\,b^4\,c\,e^5-60992\,g\,b^3\,c^2\,d^2\,e^3-11648\,f\,b^3\,c^2\,d\,e^4+91008\,g\,b^2\,c^3\,d^3\,e^2+27552\,f\,b^2\,c^3\,d^2\,e^3-66252\,g\,b\,c^4\,d^4\,e-27888\,f\,b\,c^4\,d^3\,e^2+18828\,g\,c^5\,d^5+10122\,f\,c^5\,d^4\,e\right )}{105\,c^8\,e^5}+\frac {2\,g\,x^5\,\sqrt {d+e\,x}}{7\,c^3}+\frac {4\,x^3\,\sqrt {d+e\,x}\,\left (40\,g\,b^2\,e^2-192\,g\,b\,c\,d\,e-28\,f\,b\,c\,e^2+257\,g\,c^2\,d^2+98\,f\,c^2\,d\,e\right )}{105\,c^5\,e^2}+\frac {2\,x^4\,\sqrt {d+e\,x}\,\left (38\,c\,d\,g-10\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^4\,e}-\frac {x\,\sqrt {d+e\,x}\,\left (3840\,g\,b^4\,c\,e^5-26112\,g\,b^3\,c^2\,d\,e^4-2688\,f\,b^3\,c^2\,e^5+65376\,g\,b^2\,c^3\,d^2\,e^3+14784\,f\,b^2\,c^3\,d\,e^4-71136\,g\,b\,c^4\,d^3\,e^2-26544\,f\,b\,c^4\,d^2\,e^3+28242\,g\,c^5\,d^4\,e+15288\,f\,c^5\,d^3\,e^2\right )}{105\,c^8\,e^5}+\frac {x^2\,\sqrt {d+e\,x}\,\left (-960\,g\,b^3\,c^2\,e^5+5568\,g\,b^2\,c^3\,d\,e^4+672\,f\,b^2\,c^3\,e^5-10776\,g\,b\,c^4\,d^2\,e^3-3024\,f\,b\,c^4\,d\,e^4+7008\,g\,c^5\,d^3\,e^2+3612\,f\,c^5\,d^2\,e^3\right )}{105\,c^8\,e^5}\right )}{x^3+\frac {x\,\left (105\,b^2\,c^6\,e^5-105\,c^8\,d^2\,e^3\right )}{105\,c^8\,e^5}+\frac {d\,{\left (b\,e-c\,d\right )}^2}{c^2\,e^3}+\frac {x^2\,\left (2\,b\,e-c\,d\right )}{c\,e}} \]

[In]

int(((f + g*x)*(d + e*x)^(13/2))/(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(5/2),x)

[Out]

-((c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2)*(((d + e*x)^(1/2)*(18828*c^5*d^5*g - 2560*b^5*e^5*g + 1792*b^4*c
*e^5*f + 10122*c^5*d^4*e*f - 66252*b*c^4*d^4*e*g + 19968*b^4*c*d*e^4*g - 27888*b*c^4*d^3*e^2*f - 11648*b^3*c^2
*d*e^4*f + 27552*b^2*c^3*d^2*e^3*f + 91008*b^2*c^3*d^3*e^2*g - 60992*b^3*c^2*d^2*e^3*g))/(105*c^8*e^5) + (2*g*
x^5*(d + e*x)^(1/2))/(7*c^3) + (4*x^3*(d + e*x)^(1/2)*(40*b^2*e^2*g + 257*c^2*d^2*g - 28*b*c*e^2*f + 98*c^2*d*
e*f - 192*b*c*d*e*g))/(105*c^5*e^2) + (2*x^4*(d + e*x)^(1/2)*(38*c*d*g - 10*b*e*g + 7*c*e*f))/(35*c^4*e) - (x*
(d + e*x)^(1/2)*(15288*c^5*d^3*e^2*f - 2688*b^3*c^2*e^5*f + 3840*b^4*c*e^5*g + 28242*c^5*d^4*e*g - 26544*b*c^4
*d^2*e^3*f + 14784*b^2*c^3*d*e^4*f - 71136*b*c^4*d^3*e^2*g - 26112*b^3*c^2*d*e^4*g + 65376*b^2*c^3*d^2*e^3*g))
/(105*c^8*e^5) + (x^2*(d + e*x)^(1/2)*(672*b^2*c^3*e^5*f - 960*b^3*c^2*e^5*g + 3612*c^5*d^2*e^3*f + 7008*c^5*d
^3*e^2*g - 3024*b*c^4*d*e^4*f - 10776*b*c^4*d^2*e^3*g + 5568*b^2*c^3*d*e^4*g))/(105*c^8*e^5)))/(x^3 + (x*(105*
b^2*c^6*e^5 - 105*c^8*d^2*e^3))/(105*c^8*e^5) + (d*(b*e - c*d)^2)/(c^2*e^3) + (x^2*(2*b*e - c*d))/(c*e))