3.2197 \(\int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx\)
Optimal. Leaf size=371 \[ \frac{5 (2 c d-b e)^7 (-9 b e g+2 c d g+16 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{11/2} e^2}+\frac{5 (b+2 c x) (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+2 c d g+16 c e f)}{16384 c^5 e}+\frac{5 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+2 c d g+16 c e f)}{6144 c^4 e}+\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+2 c d g+16 c e f)}{384 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2} \]
[Out]
(5*(2*c*d - b*e)^5*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2])/(16384*c^5*e) + (5*(2*c*d - b*e)^3*(16*c*e*f + 2*c*d*g
- 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(6144*c^4*e)
+ ((2*c*d - b*e)*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*
e^2*x - c*e^2*x^2)^(5/2))/(384*c^3*e) + ((9*b*e*g - 16*c*(e*f + d*g) - 14*c*e*g*
x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(112*c^2*e^2) + (5*(2*c*d - b*e)
^7*(16*c*e*f + 2*c*d*g - 9*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2])])/(32768*c^(11/2)*e^2)
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Rubi [A] time = 1.20874, antiderivative size = 371, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095
\[ \frac{5 (2 c d-b e)^7 (-9 b e g+2 c d g+16 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{11/2} e^2}+\frac{5 (b+2 c x) (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+2 c d g+16 c e f)}{16384 c^5 e}+\frac{5 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+2 c d g+16 c e f)}{6144 c^4 e}+\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+2 c d g+16 c e f)}{384 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
(5*(2*c*d - b*e)^5*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2])/(16384*c^5*e) + (5*(2*c*d - b*e)^3*(16*c*e*f + 2*c*d*g
- 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(6144*c^4*e)
+ ((2*c*d - b*e)*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*
e^2*x - c*e^2*x^2)^(5/2))/(384*c^3*e) + ((9*b*e*g - 16*c*(e*f + d*g) - 14*c*e*g*
x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(112*c^2*e^2) + (5*(2*c*d - b*e)
^7*(16*c*e*f + 2*c*d*g - 9*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d
- b*e) - b*e^2*x - c*e^2*x^2])])/(32768*c^(11/2)*e^2)
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Rubi in Sympy [A] time = 111.682, size = 401, normalized size = 1.08 \[ - \frac{g \left (d + e x\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{8 c e^{2}} + \frac{\left (\frac{9 b e g}{2} - c d g - 8 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{7}{2}}}{56 c^{2} e^{2}} + \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (9 b e g - 2 c d g - 16 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{5}{2}}}{384 c^{3} e} + \frac{5 \left (b + 2 c x\right ) \left (b e - 2 c d\right )^{3} \left (9 b e g - 2 c d g - 16 c e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{6144 c^{4} e} + \frac{5 \left (b + 2 c x\right ) \left (b e - 2 c d\right )^{5} \left (9 b e g - 2 c d g - 16 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{16384 c^{5} e} + \frac{5 \left (b e - 2 c d\right )^{7} \left (9 b e g - 2 c d g - 16 c e f\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{32768 c^{\frac{11}{2}} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
-g*(d + e*x)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(8*c*e**2) + (9*b
*e*g/2 - c*d*g - 8*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))**(7/2)/(56*
c**2*e**2) + (b + 2*c*x)*(b*e - 2*c*d)*(9*b*e*g - 2*c*d*g - 16*c*e*f)*(-b*e**2*x
- c*e**2*x**2 + d*(-b*e + c*d))**(5/2)/(384*c**3*e) + 5*(b + 2*c*x)*(b*e - 2*c*
d)**3*(9*b*e*g - 2*c*d*g - 16*c*e*f)*(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))*
*(3/2)/(6144*c**4*e) + 5*(b + 2*c*x)*(b*e - 2*c*d)**5*(9*b*e*g - 2*c*d*g - 16*c*
e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(16384*c**5*e) + 5*(b*e - 2*
c*d)**7*(9*b*e*g - 2*c*d*g - 16*c*e*f)*atan(-e*(-b - 2*c*x)/(2*sqrt(c)*sqrt(-b*e
**2*x - c*e**2*x**2 + d*(-b*e + c*d))))/(32768*c**(11/2)*e**2)
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Mathematica [C] time = 6.43598, size = 991, normalized size = 2.67 \[ \frac{\left (\frac{1}{8} c^2 e^5 g x^7+\frac{1}{112} c e^4 (16 c e f+16 c d g+33 b e g) x^6+\frac{e^3 \left (224 d e f c^2-476 d^2 g c^2+464 b e^2 f c+940 b d e g c+243 b^2 e^2 g\right ) x^5}{1344}+\frac{e^2 \left (-1152 d^2 e f c^3-1152 d^3 g c^3+2272 b d e^2 f c^2-76 b d^2 e g c^2+592 b^2 e^3 f c+1820 b^2 d e^2 g c+3 b^3 e^3 g\right ) x^4}{2688 c}+\frac{e \left (-11648 d^3 e f c^4+6608 d^4 g c^4-960 b d^2 e^2 f c^3-25824 b d^3 e g c^3+18656 b^2 d e^3 f c^2+19000 b^2 d^2 e^2 g c^2+48 b^3 e^4 f c+264 b^3 d e^3 g c-27 b^4 e^4 g\right ) x^3}{21504 c^2}+\frac{\left (18432 d^4 e f c^5+18432 d^5 g c^5-71808 b d^3 e^2 f c^4-35472 b d^4 e g c^4+52416 b^2 d^2 e^3 f c^3+14688 b^2 d^3 e^2 g c^3+1184 b^3 d e^4 f c^2+2920 b^3 d^2 e^3 g c^2-112 b^4 e^5 f c-680 b^4 d e^4 g c+63 b^5 e^5 g\right ) x^2}{43008 c^3}+\frac{\left (118272 d^5 e f c^6-6720 d^6 g c^6-221952 b d^4 e^2 f c^5+34752 b d^5 e g c^5+78336 b^2 d^3 e^3 f c^4-58896 b^2 d^4 e^2 g c^4+30720 b^3 d^2 e^4 f c^3+45792 b^3 d^3 e^3 g c^3-6496 b^4 d e^5 f c^2-18092 b^4 d^2 e^4 g c^2+560 b^5 e^6 f c+3724 b^5 d e^5 g c-315 b^6 e^6 g\right ) x}{172032 c^4 e}+\frac{-49152 d^6 e f c^7-49152 d^7 g c^7+265728 b d^5 e^2 f c^6+189888 b d^6 e g c^6-443136 b^2 d^4 e^3 f c^5-333888 b^2 d^5 e^2 g c^5+321536 b^3 d^3 e^4 f c^4+332464 b^3 d^4 e^3 g c^4-112896 b^4 d^2 e^5 f c^3-194976 b^4 d^3 e^4 g c^3+21280 b^5 d e^6 f c^2+66164 b^5 d^2 e^5 g c^2-1680 b^6 e^7 f c-12180 b^6 d e^6 g c+945 b^7 e^7 g}{344064 c^5 e^2}\right ) ((d+e x) (c (d-e x)-b e))^{5/2}}{(d+e x)^2 (c d-b e-c e x)^2}-\frac{5 i (b e-2 c d)^7 (16 c e f+2 c d g-9 b e g) ((d+e x) (c (d-e x)-b e))^{5/2} \log \left (2 \sqrt{d+e x} \sqrt{c d-b e-c e x}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{32768 c^{11/2} e^2 (d+e x)^{5/2} (c d-b e-c e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]
[Out]
(((-49152*c^7*d^6*e*f + 265728*b*c^6*d^5*e^2*f - 443136*b^2*c^5*d^4*e^3*f + 3215
36*b^3*c^4*d^3*e^4*f - 112896*b^4*c^3*d^2*e^5*f + 21280*b^5*c^2*d*e^6*f - 1680*b
^6*c*e^7*f - 49152*c^7*d^7*g + 189888*b*c^6*d^6*e*g - 333888*b^2*c^5*d^5*e^2*g +
332464*b^3*c^4*d^4*e^3*g - 194976*b^4*c^3*d^3*e^4*g + 66164*b^5*c^2*d^2*e^5*g -
12180*b^6*c*d*e^6*g + 945*b^7*e^7*g)/(344064*c^5*e^2) + ((118272*c^6*d^5*e*f -
221952*b*c^5*d^4*e^2*f + 78336*b^2*c^4*d^3*e^3*f + 30720*b^3*c^3*d^2*e^4*f - 649
6*b^4*c^2*d*e^5*f + 560*b^5*c*e^6*f - 6720*c^6*d^6*g + 34752*b*c^5*d^5*e*g - 588
96*b^2*c^4*d^4*e^2*g + 45792*b^3*c^3*d^3*e^3*g - 18092*b^4*c^2*d^2*e^4*g + 3724*
b^5*c*d*e^5*g - 315*b^6*e^6*g)*x)/(172032*c^4*e) + ((18432*c^5*d^4*e*f - 71808*b
*c^4*d^3*e^2*f + 52416*b^2*c^3*d^2*e^3*f + 1184*b^3*c^2*d*e^4*f - 112*b^4*c*e^5*
f + 18432*c^5*d^5*g - 35472*b*c^4*d^4*e*g + 14688*b^2*c^3*d^3*e^2*g + 2920*b^3*c
^2*d^2*e^3*g - 680*b^4*c*d*e^4*g + 63*b^5*e^5*g)*x^2)/(43008*c^3) + (e*(-11648*c
^4*d^3*e*f - 960*b*c^3*d^2*e^2*f + 18656*b^2*c^2*d*e^3*f + 48*b^3*c*e^4*f + 6608
*c^4*d^4*g - 25824*b*c^3*d^3*e*g + 19000*b^2*c^2*d^2*e^2*g + 264*b^3*c*d*e^3*g -
27*b^4*e^4*g)*x^3)/(21504*c^2) + (e^2*(-1152*c^3*d^2*e*f + 2272*b*c^2*d*e^2*f +
592*b^2*c*e^3*f - 1152*c^3*d^3*g - 76*b*c^2*d^2*e*g + 1820*b^2*c*d*e^2*g + 3*b^
3*e^3*g)*x^4)/(2688*c) + (e^3*(224*c^2*d*e*f + 464*b*c*e^2*f - 476*c^2*d^2*g + 9
40*b*c*d*e*g + 243*b^2*e^2*g)*x^5)/1344 + (c*e^4*(16*c*e*f + 16*c*d*g + 33*b*e*g
)*x^6)/112 + (c^2*e^5*g*x^7)/8)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))/((d +
e*x)^2*(c*d - b*e - c*e*x)^2) - (((5*I)/32768)*(-2*c*d + b*e)^7*(16*c*e*f + 2*c*
d*g - 9*b*e*g)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*Log[((-I)*e*(b + 2*c*x))
/Sqrt[c] + 2*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]])/(c^(11/2)*e^2*(d + e*x)^(5/
2)*(c*d - b*e - c*e*x)^(5/2))
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Maple [B] time = 0.02, size = 3472, normalized size = 9.4 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)
[Out]
45/32768*e^7*g*b^8/c^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(1/2))+15/1024*e^3*g*b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(3/2)*x+45/8192*e^5*g*b^6/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+3/64
*e*g*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x+5/128/e*g*c^2*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^6+5/192/e*g*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3
/2)*x*d^4+5/256/e*g*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^6+475/4096*b^5/
c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*g-125/512*b^4/c^2*e^2*(-c*e^2
*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*g+25/256*b^3/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(3/2)*d^2*g+475/2048*b^4/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*
x*d^2*g+105/64*f*d^5*c*e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^
2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^2+175/256*f*d^3/c*e^4/(c*e^2)^(1/2)*arctan((
c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^4-25/64*f*d^2
/c*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^3-5/256*b^7/c^4*e^6/(c*e^2)^(1
/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d*g
-35/384*b^3/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d*g-35/32*b^3*e^2/(
c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(
1/2))*d^5*g-5/48*b/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*d*g-1/12*b/c*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*e*f-35/768*b^4/c^3*e^2*(-c*e^2*x^2-b*e^2*x-b
*d*e+c*d^2)^(3/2)*d*g-115/4096*b^6/c^4*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2
)*d*g-5/16*b*c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2))*d^7*g+275/512*b^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/
2)*x*d^4*g+25/256*f*d/c^2*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^4-25/12
8*f*d^2/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^4+25/64*f*d^3/c*e^2*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^3-5/32*f*d^2/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(3/2)*b^2-5/16*f*d^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b+25/32*
f*d^3*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b^2-175/128*f*d^4*e^3/(c*e^2)
^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*
b^3+5/64*f*d/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^3+25/512*f*d/c^3*e
^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^5+5/128/e*g*c^3/(c*e^2)^(1/2)*arctan
((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^8+1/96/e*g/
c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b*d^2-5/32*b*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(3/2)*x*d^3*g+5/24*f*d^3*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x+5/32*
f*d^5*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b+5/16*f*d^7*c^3/(c*e^2)^(1/2)*ar
ctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))+9/112/e*g
*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)+15/2048*e^3*g*b^5/c^4*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(3/2)+45/16384*e^5*g*b^7/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(1/2)-1/8/e*g*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c+5/384/e*g*(-c*e^2*x^
2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^4+3/128*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c
*d^2)^(5/2)+1/48/e*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*d^2+1/12*f*d/c*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b-25/64*f*d^4*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(1/2)*b^2-5/64*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^3*g-5/384*b^4
/c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f-5/1024*b^6/c^4*e^5*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-5/96*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2
)*d*g-1/24*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*e*f-125/256*b^3/c*e^2*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3*g+875/1024*b^4/c*e^3/(c*e^2)^(1/2)*
arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^4*g+2
5/128*b^2/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2*g-1/7*(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(7/2)/c/e*f-35/256*b^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*
d^5*g+1/6*f*d*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x-25/32*f*d^4*c*e*(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*b+35/1024*f*d/c^3*e^6/(c*e^2)^(1/2)*arctan((c*e^
2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^6-115/2048*b^5/c^
3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d*g+245/2048*b^6/c^3*e^5/(c*e^2)^
(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d
^2*g-105/256*b^5/c^2*e^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^3*g+105/128*b^2*c*e/(c*e^2)^(1/2)*arctan((c*e^
2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^6*g+5/16*f*d^5*c^
2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+5/48*f*d^3*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(3/2)*b-5/2048*b^7/c^4*e^7/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)
/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*f+275/1024*b^3/c*e*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2)*d^4*g-5/512*b^5/c^3*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2
)*x*f-35/128*b*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5*g-5/192*b^3/c^2*e^
3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*f-1/7*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(7/2)/c/e^2*d*g-35/32*f*d^6*c^2*e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/
c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b-105/512*f*d^2/c^2*e^5/(c*e^2)^(1/2)
*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*b^5+5/
32*f*d/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*b^2
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)*(g*x + f),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 11.0021, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)*(g*x + f),x, algorithm="fricas")
[Out]
[1/1376256*(4*(43008*c^7*e^7*g*x^7 + 3072*(16*c^7*e^7*f + (16*c^7*d*e^6 + 33*b*c
^6*e^7)*g)*x^6 + 256*(16*(14*c^7*d*e^6 + 29*b*c^6*e^7)*f - (476*c^7*d^2*e^5 - 94
0*b*c^6*d*e^6 - 243*b^2*c^5*e^7)*g)*x^5 - 128*(16*(72*c^7*d^2*e^5 - 142*b*c^6*d*
e^6 - 37*b^2*c^5*e^7)*f + (1152*c^7*d^3*e^4 + 76*b*c^6*d^2*e^5 - 1820*b^2*c^5*d*
e^6 - 3*b^3*c^4*e^7)*g)*x^4 - 16*(16*(728*c^7*d^3*e^4 + 60*b*c^6*d^2*e^5 - 1166*
b^2*c^5*d*e^6 - 3*b^3*c^4*e^7)*f - (6608*c^7*d^4*e^3 - 25824*b*c^6*d^3*e^4 + 190
00*b^2*c^5*d^2*e^5 + 264*b^3*c^4*d*e^6 - 27*b^4*c^3*e^7)*g)*x^3 + 8*(16*(1152*c^
7*d^4*e^3 - 4488*b*c^6*d^3*e^4 + 3276*b^2*c^5*d^2*e^5 + 74*b^3*c^4*d*e^6 - 7*b^4
*c^3*e^7)*f + (18432*c^7*d^5*e^2 - 35472*b*c^6*d^4*e^3 + 14688*b^2*c^5*d^3*e^4 +
2920*b^3*c^4*d^2*e^5 - 680*b^4*c^3*d*e^6 + 63*b^5*c^2*e^7)*g)*x^2 - 16*(3072*c^
7*d^6*e - 16608*b*c^6*d^5*e^2 + 27696*b^2*c^5*d^4*e^3 - 20096*b^3*c^4*d^3*e^4 +
7056*b^4*c^3*d^2*e^5 - 1330*b^5*c^2*d*e^6 + 105*b^6*c*e^7)*f - (49152*c^7*d^7 -
189888*b*c^6*d^6*e + 333888*b^2*c^5*d^5*e^2 - 332464*b^3*c^4*d^4*e^3 + 194976*b^
4*c^3*d^3*e^4 - 66164*b^5*c^2*d^2*e^5 + 12180*b^6*c*d*e^6 - 945*b^7*e^7)*g + 2*(
16*(7392*c^7*d^5*e^2 - 13872*b*c^6*d^4*e^3 + 4896*b^2*c^5*d^3*e^4 + 1920*b^3*c^4
*d^2*e^5 - 406*b^4*c^3*d*e^6 + 35*b^5*c^2*e^7)*f - (6720*c^7*d^6*e - 34752*b*c^6
*d^5*e^2 + 58896*b^2*c^5*d^4*e^3 - 45792*b^3*c^4*d^3*e^4 + 18092*b^4*c^3*d^2*e^5
- 3724*b^5*c^2*d*e^6 + 315*b^6*c*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 -
b*d*e)*sqrt(-c) + 105*(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^6*d^5*
e^3 - 560*b^3*c^5*d^4*e^4 + 280*b^4*c^4*d^3*e^5 - 84*b^5*c^3*d^2*e^6 + 14*b^6*c^
2*d*e^7 - b^7*c*e^8)*f + (256*c^8*d^8 - 2048*b*c^7*d^7*e + 5376*b^2*c^6*d^6*e^2
- 7168*b^3*c^5*d^5*e^3 + 5600*b^4*c^4*d^4*e^4 - 2688*b^5*c^3*d^3*e^5 + 784*b^6*c
^2*d^2*e^6 - 128*b^7*c*d*e^7 + 9*b^8*e^8)*g)*log(4*sqrt(-c*e^2*x^2 - b*e^2*x + c
*d^2 - b*d*e)*(2*c^2*e*x + b*c*e) + (8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4
*b*c*d*e + b^2*e^2)*sqrt(-c)))/(sqrt(-c)*c^5*e^2), 1/688128*(2*(43008*c^7*e^7*g*
x^7 + 3072*(16*c^7*e^7*f + (16*c^7*d*e^6 + 33*b*c^6*e^7)*g)*x^6 + 256*(16*(14*c^
7*d*e^6 + 29*b*c^6*e^7)*f - (476*c^7*d^2*e^5 - 940*b*c^6*d*e^6 - 243*b^2*c^5*e^7
)*g)*x^5 - 128*(16*(72*c^7*d^2*e^5 - 142*b*c^6*d*e^6 - 37*b^2*c^5*e^7)*f + (1152
*c^7*d^3*e^4 + 76*b*c^6*d^2*e^5 - 1820*b^2*c^5*d*e^6 - 3*b^3*c^4*e^7)*g)*x^4 - 1
6*(16*(728*c^7*d^3*e^4 + 60*b*c^6*d^2*e^5 - 1166*b^2*c^5*d*e^6 - 3*b^3*c^4*e^7)*
f - (6608*c^7*d^4*e^3 - 25824*b*c^6*d^3*e^4 + 19000*b^2*c^5*d^2*e^5 + 264*b^3*c^
4*d*e^6 - 27*b^4*c^3*e^7)*g)*x^3 + 8*(16*(1152*c^7*d^4*e^3 - 4488*b*c^6*d^3*e^4
+ 3276*b^2*c^5*d^2*e^5 + 74*b^3*c^4*d*e^6 - 7*b^4*c^3*e^7)*f + (18432*c^7*d^5*e^
2 - 35472*b*c^6*d^4*e^3 + 14688*b^2*c^5*d^3*e^4 + 2920*b^3*c^4*d^2*e^5 - 680*b^4
*c^3*d*e^6 + 63*b^5*c^2*e^7)*g)*x^2 - 16*(3072*c^7*d^6*e - 16608*b*c^6*d^5*e^2 +
27696*b^2*c^5*d^4*e^3 - 20096*b^3*c^4*d^3*e^4 + 7056*b^4*c^3*d^2*e^5 - 1330*b^5
*c^2*d*e^6 + 105*b^6*c*e^7)*f - (49152*c^7*d^7 - 189888*b*c^6*d^6*e + 333888*b^2
*c^5*d^5*e^2 - 332464*b^3*c^4*d^4*e^3 + 194976*b^4*c^3*d^3*e^4 - 66164*b^5*c^2*d
^2*e^5 + 12180*b^6*c*d*e^6 - 945*b^7*e^7)*g + 2*(16*(7392*c^7*d^5*e^2 - 13872*b*
c^6*d^4*e^3 + 4896*b^2*c^5*d^3*e^4 + 1920*b^3*c^4*d^2*e^5 - 406*b^4*c^3*d*e^6 +
35*b^5*c^2*e^7)*f - (6720*c^7*d^6*e - 34752*b*c^6*d^5*e^2 + 58896*b^2*c^5*d^4*e^
3 - 45792*b^3*c^4*d^3*e^4 + 18092*b^4*c^3*d^2*e^5 - 3724*b^5*c^2*d*e^6 + 315*b^6
*c*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(c) + 105*(16*(128*
c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^6*d^5*e^3 - 560*b^3*c^5*d^4*e^4 + 280*
b^4*c^4*d^3*e^5 - 84*b^5*c^3*d^2*e^6 + 14*b^6*c^2*d*e^7 - b^7*c*e^8)*f + (256*c^
8*d^8 - 2048*b*c^7*d^7*e + 5376*b^2*c^6*d^6*e^2 - 7168*b^3*c^5*d^5*e^3 + 5600*b^
4*c^4*d^4*e^4 - 2688*b^5*c^3*d^3*e^5 + 784*b^6*c^2*d^2*e^6 - 128*b^7*c*d*e^7 + 9
*b^8*e^8)*g)*arctan(1/2*(2*c*e*x + b*e)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d
*e)*sqrt(c))))/(c^(11/2)*e^2)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}} \left (d + e x\right ) \left (f + g x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(d + e*x)*(f + g*x), x)
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GIAC/XCAS [A] time = 0.331986, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(5/2)*(e*x + d)*(g*x + f),x, algorithm="giac")
[Out]
Done