3.302 \(\int (d+e x)^3 \left (b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=332 \[ -\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{12288 c^5}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{768 c^4}+\frac{e \left (b x+c x^2\right )^{7/2} \left (99 b^2 e^2+154 c e x (2 c d-b e)-486 b c d e+640 c^2 d^2\right )}{2016 c^3}-\frac{5 b^6 (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{32768 c^{13/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{32768 c^6}+\frac{e \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

[Out]

(5*b^4*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x
+ c*x^2])/(32768*c^6) - (5*b^2*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*
e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(12288*c^5) + ((2*c*d - b*e)*(32*c^2*d^2 -
32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e
*x)^2*(b*x + c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^2 - 486*b*c*d*e + 99*b^2*e^2 +
154*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(7/2))/(2016*c^3) - (5*b^6*(2*c*d - b*e)*
(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(
32768*c^(13/2))

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Rubi [A]  time = 0.983768, antiderivative size = 332, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{5 b^2 (b+2 c x) \left (b x+c x^2\right )^{3/2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{12288 c^5}+\frac{(b+2 c x) \left (b x+c x^2\right )^{5/2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{768 c^4}+\frac{e \left (b x+c x^2\right )^{7/2} \left (99 b^2 e^2+154 c e x (2 c d-b e)-486 b c d e+640 c^2 d^2\right )}{2016 c^3}-\frac{5 b^6 (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{32768 c^{13/2}}+\frac{5 b^4 (b+2 c x) \sqrt{b x+c x^2} (2 c d-b e) \left (11 b^2 e^2-32 b c d e+32 c^2 d^2\right )}{32768 c^6}+\frac{e \left (b x+c x^2\right )^{7/2} (d+e x)^2}{9 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(b*x + c*x^2)^(5/2),x]

[Out]

(5*b^4*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*Sqrt[b*x
+ c*x^2])/(32768*c^6) - (5*b^2*(2*c*d - b*e)*(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*
e^2)*(b + 2*c*x)*(b*x + c*x^2)^(3/2))/(12288*c^5) + ((2*c*d - b*e)*(32*c^2*d^2 -
32*b*c*d*e + 11*b^2*e^2)*(b + 2*c*x)*(b*x + c*x^2)^(5/2))/(768*c^4) + (e*(d + e
*x)^2*(b*x + c*x^2)^(7/2))/(9*c) + (e*(640*c^2*d^2 - 486*b*c*d*e + 99*b^2*e^2 +
154*c*e*(2*c*d - b*e)*x)*(b*x + c*x^2)^(7/2))/(2016*c^3) - (5*b^6*(2*c*d - b*e)*
(32*c^2*d^2 - 32*b*c*d*e + 11*b^2*e^2)*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(
32768*c^(13/2))

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Rubi in Sympy [A]  time = 57.8273, size = 337, normalized size = 1.02 \[ \frac{5 b^{6} \left (b e - 2 c d\right ) \left (11 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{32768 c^{\frac{13}{2}}} - \frac{5 b^{4} \left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}} \left (11 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{32768 c^{6}} + \frac{5 b^{2} \left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (11 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{12288 c^{5}} + \frac{e \left (d + e x\right )^{2} \left (b x + c x^{2}\right )^{\frac{7}{2}}}{9 c} + \frac{e \left (b x + c x^{2}\right )^{\frac{7}{2}} \left (\frac{99 b^{2} e^{2}}{4} - \frac{243 b c d e}{2} + 160 c^{2} d^{2} - \frac{77 c e x \left (b e - 2 c d\right )}{2}\right )}{504 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \left (b x + c x^{2}\right )^{\frac{5}{2}} \left (11 b^{2} e^{2} - 32 b c d e + 32 c^{2} d^{2}\right )}{768 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(c*x**2+b*x)**(5/2),x)

[Out]

5*b**6*(b*e - 2*c*d)*(11*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)*atanh(sqrt(c)*x/
sqrt(b*x + c*x**2))/(32768*c**(13/2)) - 5*b**4*(b + 2*c*x)*(b*e - 2*c*d)*sqrt(b*
x + c*x**2)*(11*b**2*e**2 - 32*b*c*d*e + 32*c**2*d**2)/(32768*c**6) + 5*b**2*(b
+ 2*c*x)*(b*e - 2*c*d)*(b*x + c*x**2)**(3/2)*(11*b**2*e**2 - 32*b*c*d*e + 32*c**
2*d**2)/(12288*c**5) + e*(d + e*x)**2*(b*x + c*x**2)**(7/2)/(9*c) + e*(b*x + c*x
**2)**(7/2)*(99*b**2*e**2/4 - 243*b*c*d*e/2 + 160*c**2*d**2 - 77*c*e*x*(b*e - 2*
c*d)/2)/(504*c**3) - (b + 2*c*x)*(b*e - 2*c*d)*(b*x + c*x**2)**(5/2)*(11*b**2*e*
*2 - 32*b*c*d*e + 32*c**2*d**2)/(768*c**4)

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Mathematica [A]  time = 0.804266, size = 397, normalized size = 1.2 \[ \frac{\sqrt{x (b+c x)} \left (\frac{315 b^6 \left (11 b^3 e^3-54 b^2 c d e^2+96 b c^2 d^2 e-64 c^3 d^3\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (-3465 b^8 e^3+210 b^7 c e^2 (81 d+11 e x)-84 b^6 c^2 e \left (360 d^2+135 d e x+22 e^2 x^2\right )+144 b^5 c^3 \left (140 d^3+140 d^2 e x+63 d e^2 x^2+11 e^3 x^3\right )-32 b^4 c^4 x \left (420 d^3+504 d^2 e x+243 d e^2 x^2+44 e^3 x^3\right )+256 b^3 c^5 x^2 \left (42 d^3+54 d^2 e x+27 d e^2 x^2+5 e^3 x^3\right )+1536 b^2 c^6 x^3 \left (378 d^3+888 d^2 e x+729 d e^2 x^2+206 e^3 x^3\right )+2048 b c^7 x^4 \left (420 d^3+1044 d^2 e x+891 d e^2 x^2+259 e^3 x^3\right )+4096 c^8 x^5 \left (84 d^3+216 d^2 e x+189 d e^2 x^2+56 e^3 x^3\right )\right )\right )}{2064384 c^{13/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(b*x + c*x^2)^(5/2),x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(-3465*b^8*e^3 + 210*b^7*c*e^2*(81*d + 11*e*x) - 84*
b^6*c^2*e*(360*d^2 + 135*d*e*x + 22*e^2*x^2) + 256*b^3*c^5*x^2*(42*d^3 + 54*d^2*
e*x + 27*d*e^2*x^2 + 5*e^3*x^3) + 144*b^5*c^3*(140*d^3 + 140*d^2*e*x + 63*d*e^2*
x^2 + 11*e^3*x^3) - 32*b^4*c^4*x*(420*d^3 + 504*d^2*e*x + 243*d*e^2*x^2 + 44*e^3
*x^3) + 4096*c^8*x^5*(84*d^3 + 216*d^2*e*x + 189*d*e^2*x^2 + 56*e^3*x^3) + 1536*
b^2*c^6*x^3*(378*d^3 + 888*d^2*e*x + 729*d*e^2*x^2 + 206*e^3*x^3) + 2048*b*c^7*x
^4*(420*d^3 + 1044*d^2*e*x + 891*d*e^2*x^2 + 259*e^3*x^3)) + (315*b^6*(-64*c^3*d
^3 + 96*b*c^2*d^2*e - 54*b^2*c*d*e^2 + 11*b^3*e^3)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[
b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(2064384*c^(13/2))

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Maple [B]  time = 0.021, size = 813, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(c*x^2+b*x)^(5/2),x)

[Out]

5/64*d^2*e*b^3/c^2*(c*x^2+b*x)^(3/2)*x-15/512*d^2*e*b^5/c^3*(c*x^2+b*x)^(1/2)*x-
5/1024*d^3*b^6/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+3/7*d^2*e*(c*x^
2+b*x)^(7/2)/c+1/12*d^3/c*(c*x^2+b*x)^(5/2)*b-5/192*d^3*b^3/c^2*(c*x^2+b*x)^(3/2
)+5/512*d^3*b^5/c^3*(c*x^2+b*x)^(1/2)+1/9*e^3*x^2*(c*x^2+b*x)^(7/2)/c+11/224*e^3
*b^2/c^3*(c*x^2+b*x)^(7/2)-11/768*e^3*b^4/c^4*(c*x^2+b*x)^(5/2)+55/12288*e^3*b^6
/c^5*(c*x^2+b*x)^(3/2)-55/32768*e^3*b^8/c^6*(c*x^2+b*x)^(1/2)+55/65536*e^3*b^9/c
^(13/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+9/64*d*e^2*b^2/c^2*(c*x^2+b*x)
^(5/2)*x-45/1024*d*e^2*b^4/c^3*(c*x^2+b*x)^(3/2)*x+135/8192*d*e^2*b^6/c^4*(c*x^2
+b*x)^(1/2)*x-1/4*d^2*e*b/c*(c*x^2+b*x)^(5/2)*x+5/128*d^2*e*b^4/c^3*(c*x^2+b*x)^
(3/2)-15/1024*d^2*e*b^6/c^4*(c*x^2+b*x)^(1/2)+15/2048*d^2*e*b^7/c^(9/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+9/128*d*e^2*b^3/c^3*(c*x^2+b*x)^(5/2)-45/2048*
d*e^2*b^5/c^4*(c*x^2+b*x)^(3/2)+135/16384*d*e^2*b^7/c^5*(c*x^2+b*x)^(1/2)-11/384
*e^3*b^3/c^3*(c*x^2+b*x)^(5/2)*x+55/6144*e^3*b^5/c^4*(c*x^2+b*x)^(3/2)*x-55/1638
4*e^3*b^7/c^5*(c*x^2+b*x)^(1/2)*x-11/144*e^3*b/c^2*x*(c*x^2+b*x)^(7/2)-1/8*d^2*e
*b^2/c^2*(c*x^2+b*x)^(5/2)+1/6*d^3*(c*x^2+b*x)^(5/2)*x-135/32768*d*e^2*b^8/c^(11
/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2))+3/8*d*e^2*x*(c*x^2+b*x)^(7/2)/c-27
/112*d*e^2*b/c^2*(c*x^2+b*x)^(7/2)+5/256*d^3*b^4/c^2*(c*x^2+b*x)^(1/2)*x-5/96*d^
3*b^2/c*(c*x^2+b*x)^(3/2)*x

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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*x^2 + b*x)^(5/2)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264555, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/4128768*(2*(229376*c^8*e^3*x^8 + 20160*b^5*c^3*d^3 - 30240*b^6*c^2*d^2*e + 17
010*b^7*c*d*e^2 - 3465*b^8*e^3 + 14336*(54*c^8*d*e^2 + 37*b*c^7*e^3)*x^7 + 3072*
(288*c^8*d^2*e + 594*b*c^7*d*e^2 + 103*b^2*c^6*e^3)*x^6 + 256*(1344*c^8*d^3 + 83
52*b*c^7*d^2*e + 4374*b^2*c^6*d*e^2 + 5*b^3*c^5*e^3)*x^5 + 128*(6720*b*c^7*d^3 +
10656*b^2*c^6*d^2*e + 54*b^3*c^5*d*e^2 - 11*b^4*c^4*e^3)*x^4 + 144*(4032*b^2*c^
6*d^3 + 96*b^3*c^5*d^2*e - 54*b^4*c^4*d*e^2 + 11*b^5*c^3*e^3)*x^3 + 168*(64*b^3*
c^5*d^3 - 96*b^4*c^4*d^2*e + 54*b^5*c^3*d*e^2 - 11*b^6*c^2*e^3)*x^2 - 210*(64*b^
4*c^4*d^3 - 96*b^5*c^3*d^2*e + 54*b^6*c^2*d*e^2 - 11*b^7*c*e^3)*x)*sqrt(c*x^2 +
b*x)*sqrt(c) - 315*(64*b^6*c^3*d^3 - 96*b^7*c^2*d^2*e + 54*b^8*c*d*e^2 - 11*b^9*
e^3)*log((2*c*x + b)*sqrt(c) + 2*sqrt(c*x^2 + b*x)*c))/c^(13/2), 1/2064384*((229
376*c^8*e^3*x^8 + 20160*b^5*c^3*d^3 - 30240*b^6*c^2*d^2*e + 17010*b^7*c*d*e^2 -
3465*b^8*e^3 + 14336*(54*c^8*d*e^2 + 37*b*c^7*e^3)*x^7 + 3072*(288*c^8*d^2*e + 5
94*b*c^7*d*e^2 + 103*b^2*c^6*e^3)*x^6 + 256*(1344*c^8*d^3 + 8352*b*c^7*d^2*e + 4
374*b^2*c^6*d*e^2 + 5*b^3*c^5*e^3)*x^5 + 128*(6720*b*c^7*d^3 + 10656*b^2*c^6*d^2
*e + 54*b^3*c^5*d*e^2 - 11*b^4*c^4*e^3)*x^4 + 144*(4032*b^2*c^6*d^3 + 96*b^3*c^5
*d^2*e - 54*b^4*c^4*d*e^2 + 11*b^5*c^3*e^3)*x^3 + 168*(64*b^3*c^5*d^3 - 96*b^4*c
^4*d^2*e + 54*b^5*c^3*d*e^2 - 11*b^6*c^2*e^3)*x^2 - 210*(64*b^4*c^4*d^3 - 96*b^5
*c^3*d^2*e + 54*b^6*c^2*d*e^2 - 11*b^7*c*e^3)*x)*sqrt(c*x^2 + b*x)*sqrt(-c) - 31
5*(64*b^6*c^3*d^3 - 96*b^7*c^2*d^2*e + 54*b^8*c*d*e^2 - 11*b^9*e^3)*arctan(sqrt(
c*x^2 + b*x)*sqrt(-c)/(c*x)))/(sqrt(-c)*c^6)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(c*x**2+b*x)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(5/2)*(d + e*x)**3, x)

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GIAC/XCAS [A]  time = 0.236655, size = 648, normalized size = 1.95 \[ \frac{1}{2064384} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (4 \,{\left (14 \,{\left (16 \, c^{2} x e^{3} + \frac{54 \, c^{10} d e^{2} + 37 \, b c^{9} e^{3}}{c^{8}}\right )} x + \frac{3 \,{\left (288 \, c^{10} d^{2} e + 594 \, b c^{9} d e^{2} + 103 \, b^{2} c^{8} e^{3}\right )}}{c^{8}}\right )} x + \frac{1344 \, c^{10} d^{3} + 8352 \, b c^{9} d^{2} e + 4374 \, b^{2} c^{8} d e^{2} + 5 \, b^{3} c^{7} e^{3}}{c^{8}}\right )} x + \frac{6720 \, b c^{9} d^{3} + 10656 \, b^{2} c^{8} d^{2} e + 54 \, b^{3} c^{7} d e^{2} - 11 \, b^{4} c^{6} e^{3}}{c^{8}}\right )} x + \frac{9 \,{\left (4032 \, b^{2} c^{8} d^{3} + 96 \, b^{3} c^{7} d^{2} e - 54 \, b^{4} c^{6} d e^{2} + 11 \, b^{5} c^{5} e^{3}\right )}}{c^{8}}\right )} x + \frac{21 \,{\left (64 \, b^{3} c^{7} d^{3} - 96 \, b^{4} c^{6} d^{2} e + 54 \, b^{5} c^{5} d e^{2} - 11 \, b^{6} c^{4} e^{3}\right )}}{c^{8}}\right )} x - \frac{105 \,{\left (64 \, b^{4} c^{6} d^{3} - 96 \, b^{5} c^{5} d^{2} e + 54 \, b^{6} c^{4} d e^{2} - 11 \, b^{7} c^{3} e^{3}\right )}}{c^{8}}\right )} x + \frac{315 \,{\left (64 \, b^{5} c^{5} d^{3} - 96 \, b^{6} c^{4} d^{2} e + 54 \, b^{7} c^{3} d e^{2} - 11 \, b^{8} c^{2} e^{3}\right )}}{c^{8}}\right )} + \frac{5 \,{\left (64 \, b^{6} c^{3} d^{3} - 96 \, b^{7} c^{2} d^{2} e + 54 \, b^{8} c d e^{2} - 11 \, b^{9} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{65536 \, c^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(5/2)*(e*x + d)^3,x, algorithm="giac")

[Out]

1/2064384*sqrt(c*x^2 + b*x)*(2*(4*(2*(8*(2*(4*(14*(16*c^2*x*e^3 + (54*c^10*d*e^2
+ 37*b*c^9*e^3)/c^8)*x + 3*(288*c^10*d^2*e + 594*b*c^9*d*e^2 + 103*b^2*c^8*e^3)
/c^8)*x + (1344*c^10*d^3 + 8352*b*c^9*d^2*e + 4374*b^2*c^8*d*e^2 + 5*b^3*c^7*e^3
)/c^8)*x + (6720*b*c^9*d^3 + 10656*b^2*c^8*d^2*e + 54*b^3*c^7*d*e^2 - 11*b^4*c^6
*e^3)/c^8)*x + 9*(4032*b^2*c^8*d^3 + 96*b^3*c^7*d^2*e - 54*b^4*c^6*d*e^2 + 11*b^
5*c^5*e^3)/c^8)*x + 21*(64*b^3*c^7*d^3 - 96*b^4*c^6*d^2*e + 54*b^5*c^5*d*e^2 - 1
1*b^6*c^4*e^3)/c^8)*x - 105*(64*b^4*c^6*d^3 - 96*b^5*c^5*d^2*e + 54*b^6*c^4*d*e^
2 - 11*b^7*c^3*e^3)/c^8)*x + 315*(64*b^5*c^5*d^3 - 96*b^6*c^4*d^2*e + 54*b^7*c^3
*d*e^2 - 11*b^8*c^2*e^3)/c^8) + 5/65536*(64*b^6*c^3*d^3 - 96*b^7*c^2*d^2*e + 54*
b^8*c*d*e^2 - 11*b^9*e^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b)
)/c^(13/2)