3.938 \(\int x (A+B x) \left (a+b x+c x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=252 \[ -\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a B c-16 A b c+9 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{6144 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{384 c^3}-\frac{\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]

[Out]

(5*(b^2 - 4*a*c)^2*(9*b^2*B - 16*A*b*c - 4*a*B*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x
^2])/(16384*c^5) - (5*(b^2 - 4*a*c)*(9*b^2*B - 16*A*b*c - 4*a*B*c)*(b + 2*c*x)*(
a + b*x + c*x^2)^(3/2))/(6144*c^4) + ((9*b^2*B - 16*A*b*c - 4*a*B*c)*(b + 2*c*x)
*(a + b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B - 16*A*c - 14*B*c*x)*(a + b*x + c*
x^2)^(7/2))/(112*c^2) - (5*(b^2 - 4*a*c)^3*(9*b^2*B - 16*A*b*c - 4*a*B*c)*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(32768*c^(11/2))

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Rubi [A]  time = 0.289101, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{5 \left (b^2-4 a c\right )^3 \left (-4 a B c-16 A b c+9 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{32768 c^{11/2}}+\frac{5 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{16384 c^5}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{6144 c^4}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2} \left (-4 a B c-16 A b c+9 b^2 B\right )}{384 c^3}-\frac{\left (a+b x+c x^2\right )^{7/2} (-16 A c+9 b B-14 B c x)}{112 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(5*(b^2 - 4*a*c)^2*(9*b^2*B - 16*A*b*c - 4*a*B*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x
^2])/(16384*c^5) - (5*(b^2 - 4*a*c)*(9*b^2*B - 16*A*b*c - 4*a*B*c)*(b + 2*c*x)*(
a + b*x + c*x^2)^(3/2))/(6144*c^4) + ((9*b^2*B - 16*A*b*c - 4*a*B*c)*(b + 2*c*x)
*(a + b*x + c*x^2)^(5/2))/(384*c^3) - ((9*b*B - 16*A*c - 14*B*c*x)*(a + b*x + c*
x^2)^(7/2))/(112*c^2) - (5*(b^2 - 4*a*c)^3*(9*b^2*B - 16*A*b*c - 4*a*B*c)*ArcTan
h[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(32768*c^(11/2))

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Rubi in Sympy [A]  time = 31.8784, size = 264, normalized size = 1.05 \[ \frac{\left (a + b x + c x^{2}\right )^{\frac{7}{2}} \left (8 A c - \frac{9 B b}{2} + 7 B c x\right )}{56 c^{2}} + \frac{\left (b + 2 c x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}} \left (- 16 A b c - 4 B a c + 9 B b^{2}\right )}{384 c^{3}} - \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 16 A b c - 4 B a c + 9 B b^{2}\right )}{6144 c^{4}} + \frac{5 \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (- 16 A b c - 4 B a c + 9 B b^{2}\right )}{16384 c^{5}} - \frac{5 \left (- 4 a c + b^{2}\right )^{3} \left (- 16 A b c - 4 B a c + 9 B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{32768 c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)

[Out]

(a + b*x + c*x**2)**(7/2)*(8*A*c - 9*B*b/2 + 7*B*c*x)/(56*c**2) + (b + 2*c*x)*(a
+ b*x + c*x**2)**(5/2)*(-16*A*b*c - 4*B*a*c + 9*B*b**2)/(384*c**3) - 5*(b + 2*c
*x)*(-4*a*c + b**2)*(a + b*x + c*x**2)**(3/2)*(-16*A*b*c - 4*B*a*c + 9*B*b**2)/(
6144*c**4) + 5*(b + 2*c*x)*(-4*a*c + b**2)**2*sqrt(a + b*x + c*x**2)*(-16*A*b*c
- 4*B*a*c + 9*B*b**2)/(16384*c**5) - 5*(-4*a*c + b**2)**3*(-16*A*b*c - 4*B*a*c +
9*B*b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(32768*c**(11/2
))

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Mathematica [A]  time = 0.814566, size = 390, normalized size = 1.55 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 b^3 c^2 \left (2359 a^2 B-4 a c x (168 A+71 B x)+24 c^2 x^3 (2 A+B x)\right )+32 b^2 c^3 \left (-3 a^2 (616 A+199 B x)+12 a c x^2 (20 A+9 B x)+8 c^2 x^4 (296 A+243 B x)\right )+64 b c^3 \left (-663 a^3 B+6 a^2 c x (76 A+29 B x)+8 a c^2 x^3 (394 A+307 B x)+16 c^3 x^5 (116 A+99 B x)\right )+128 c^4 \left (3 a^3 (128 A+35 B x)+2 a^2 c x^2 (576 A+413 B x)+8 a c^2 x^4 (144 A+119 B x)+48 c^3 x^6 (8 A+7 B x)\right )+28 b^5 c (2 c x (20 A+9 B x)-375 a B)+8 b^4 c^2 \left (7 a (320 A+113 B x)-2 c x^2 (56 A+27 B x)\right )-210 b^6 c (8 A+3 B x)+945 b^7 B\right )-105 \left (b^2-4 a c\right )^3 \left (-4 a B c-16 A b c+9 b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{688128 c^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(A + B*x)*(a + b*x + c*x^2)^(5/2),x]

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^7*B - 210*b^6*c*(8*A + 3*B*x) + 28*b^5*c
*(-375*a*B + 2*c*x*(20*A + 9*B*x)) + 16*b^3*c^2*(2359*a^2*B + 24*c^2*x^3*(2*A +
B*x) - 4*a*c*x*(168*A + 71*B*x)) + 8*b^4*c^2*(-2*c*x^2*(56*A + 27*B*x) + 7*a*(32
0*A + 113*B*x)) + 32*b^2*c^3*(12*a*c*x^2*(20*A + 9*B*x) - 3*a^2*(616*A + 199*B*x
) + 8*c^2*x^4*(296*A + 243*B*x)) + 64*b*c^3*(-663*a^3*B + 6*a^2*c*x*(76*A + 29*B
*x) + 16*c^3*x^5*(116*A + 99*B*x) + 8*a*c^2*x^3*(394*A + 307*B*x)) + 128*c^4*(48
*c^3*x^6*(8*A + 7*B*x) + 3*a^3*(128*A + 35*B*x) + 8*a*c^2*x^4*(144*A + 119*B*x)
+ 2*a^2*c*x^2*(576*A + 413*B*x))) - 105*(b^2 - 4*a*c)^3*(9*b^2*B - 16*A*b*c - 4*
a*B*c)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(688128*c^(11/2))

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Maple [B]  time = 0.013, size = 1034, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x+A)*(c*x^2+b*x+a)^(5/2),x)

[Out]

5/384*A*b^4/c^3*(c*x^2+b*x+a)^(3/2)-5/1024*A*b^6/c^4*(c*x^2+b*x+a)^(1/2)+5/2048*
A*b^7/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-5/128*B*a^4/c^(3/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-45/32768*B*b^8/c^(11/2)*ln((1/2*b+c*x)
/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/8*B*x*(c*x^2+b*x+a)^(7/2)/c-15/2048*B*b^5/c^4*(c
*x^2+b*x+a)^(3/2)+45/16384*B*b^7/c^5*(c*x^2+b*x+a)^(1/2)-9/112*B*b/c^2*(c*x^2+b*
x+a)^(7/2)+3/128*B*b^3/c^3*(c*x^2+b*x+a)^(5/2)-1/24*A*b^2/c^2*(c*x^2+b*x+a)^(5/2
)-5/48*A*b/c*(c*x^2+b*x+a)^(3/2)*x*a-5/32*A*b/c*(c*x^2+b*x+a)^(1/2)*x*a^2+25/384
*B*b^2/c^2*(c*x^2+b*x+a)^(3/2)*x*a+55/512*B*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x*a^2-95
/2048*B*b^4/c^3*(c*x^2+b*x+a)^(1/2)*x*a+5/64*A*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*a+1
5/128*A*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-15/512*A*b^5
/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/64*B*b^2/c^2*(c*x^2+b*x
+a)^(5/2)*x-15/1024*B*b^4/c^3*(c*x^2+b*x+a)^(3/2)*x+55/1024*B*b^3/c^3*(c*x^2+b*x
+a)^(1/2)*a^2-95/4096*B*b^5/c^4*(c*x^2+b*x+a)^(1/2)*a-1/48*B*a/c*(c*x^2+b*x+a)^(
5/2)*x-5/192*B*a^2/c*(c*x^2+b*x+a)^(3/2)*x-5/384*B*a^2/c^2*(c*x^2+b*x+a)^(3/2)*b
-5/128*B*a^3/c*(c*x^2+b*x+a)^(1/2)*x-5/256*B*a^3/c^2*(c*x^2+b*x+a)^(1/2)*b+25/76
8*B*b^3/c^3*(c*x^2+b*x+a)^(3/2)*a+45/8192*B*b^6/c^4*(c*x^2+b*x+a)^(1/2)*x-75/102
4*B*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^2-5/32*A*b/c^(3/2)
*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3+1/7*A*(c*x^2+b*x+a)^(7/2)/c+5/1
28*A*b^4/c^3*(c*x^2+b*x+a)^(1/2)*a-5/96*A*b^2/c^2*(c*x^2+b*x+a)^(3/2)*a-5/512*A*
b^5/c^3*(c*x^2+b*x+a)^(1/2)*x-5/64*A*b^2/c^2*(c*x^2+b*x+a)^(1/2)*a^2-1/96*B*a/c^
2*(c*x^2+b*x+a)^(5/2)*b+35/2048*B*b^6/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*a+15/128*B*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a^3
-1/12*A*b/c*(c*x^2+b*x+a)^(5/2)*x+5/192*A*b^3/c^2*(c*x^2+b*x+a)^(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 + a)^(5/2)*(B*x + A)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.464523, 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 + a)^(5/2)*(B*x + A)*x,x, algorithm="fricas")

[Out]

[1/1376256*(4*(43008*B*c^7*x^7 + 945*B*b^7 + 49152*A*a^3*c^4 + 3072*(33*B*b*c^6
+ 16*A*c^7)*x^6 + 256*(243*B*b^2*c^5 + 4*(119*B*a + 116*A*b)*c^6)*x^5 + 128*(3*B
*b^3*c^4 + 1152*A*a*c^6 + 4*(307*B*a*b + 148*A*b^2)*c^5)*x^4 - 192*(221*B*a^3*b
+ 308*A*a^2*b^2)*c^3 - 16*(27*B*b^4*c^3 - 16*(413*B*a^2 + 788*A*a*b)*c^5 - 24*(9
*B*a*b^2 + 2*A*b^3)*c^4)*x^3 + 112*(337*B*a^2*b^3 + 160*A*a*b^4)*c^2 + 8*(63*B*b
^5*c^2 + 18432*A*a^2*c^5 + 48*(29*B*a^2*b + 20*A*a*b^2)*c^4 - 8*(71*B*a*b^3 + 14
*A*b^4)*c^3)*x^2 - 420*(25*B*a*b^5 + 4*A*b^6)*c - 2*(315*B*b^6*c - 192*(35*B*a^3
+ 76*A*a^2*b)*c^4 + 48*(199*B*a^2*b^2 + 112*A*a*b^3)*c^3 - 28*(113*B*a*b^4 + 20
*A*b^5)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 105*(9*B*b^8 + 256*(B*a^4 + 4*A*
a^3*b)*c^4 - 768*(B*a^3*b^2 + A*a^2*b^3)*c^3 + 96*(5*B*a^2*b^4 + 2*A*a*b^5)*c^2
- 16*(7*B*a*b^6 + A*b^7)*c)*log(4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2
*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/c^(11/2), 1/688128*(2*(43008*B*c^7*x^7 +
945*B*b^7 + 49152*A*a^3*c^4 + 3072*(33*B*b*c^6 + 16*A*c^7)*x^6 + 256*(243*B*b^2
*c^5 + 4*(119*B*a + 116*A*b)*c^6)*x^5 + 128*(3*B*b^3*c^4 + 1152*A*a*c^6 + 4*(307
*B*a*b + 148*A*b^2)*c^5)*x^4 - 192*(221*B*a^3*b + 308*A*a^2*b^2)*c^3 - 16*(27*B*
b^4*c^3 - 16*(413*B*a^2 + 788*A*a*b)*c^5 - 24*(9*B*a*b^2 + 2*A*b^3)*c^4)*x^3 + 1
12*(337*B*a^2*b^3 + 160*A*a*b^4)*c^2 + 8*(63*B*b^5*c^2 + 18432*A*a^2*c^5 + 48*(2
9*B*a^2*b + 20*A*a*b^2)*c^4 - 8*(71*B*a*b^3 + 14*A*b^4)*c^3)*x^2 - 420*(25*B*a*b
^5 + 4*A*b^6)*c - 2*(315*B*b^6*c - 192*(35*B*a^3 + 76*A*a^2*b)*c^4 + 48*(199*B*a
^2*b^2 + 112*A*a*b^3)*c^3 - 28*(113*B*a*b^4 + 20*A*b^5)*c^2)*x)*sqrt(c*x^2 + b*x
+ a)*sqrt(-c) - 105*(9*B*b^8 + 256*(B*a^4 + 4*A*a^3*b)*c^4 - 768*(B*a^3*b^2 + A
*a^2*b^3)*c^3 + 96*(5*B*a^2*b^4 + 2*A*a*b^5)*c^2 - 16*(7*B*a*b^6 + A*b^7)*c)*arc
tan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^5)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(B*x+A)*(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(x*(A + B*x)*(a + b*x + c*x**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.28549, size = 713, normalized size = 2.83 \[ \frac{1}{344064} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \,{\left (12 \,{\left (14 \, B c^{2} x + \frac{33 \, B b c^{8} + 16 \, A c^{9}}{c^{7}}\right )} x + \frac{243 \, B b^{2} c^{7} + 476 \, B a c^{8} + 464 \, A b c^{8}}{c^{7}}\right )} x + \frac{3 \, B b^{3} c^{6} + 1228 \, B a b c^{7} + 592 \, A b^{2} c^{7} + 1152 \, A a c^{8}}{c^{7}}\right )} x - \frac{27 \, B b^{4} c^{5} - 216 \, B a b^{2} c^{6} - 48 \, A b^{3} c^{6} - 6608 \, B a^{2} c^{7} - 12608 \, A a b c^{7}}{c^{7}}\right )} x + \frac{63 \, B b^{5} c^{4} - 568 \, B a b^{3} c^{5} - 112 \, A b^{4} c^{5} + 1392 \, B a^{2} b c^{6} + 960 \, A a b^{2} c^{6} + 18432 \, A a^{2} c^{7}}{c^{7}}\right )} x - \frac{315 \, B b^{6} c^{3} - 3164 \, B a b^{4} c^{4} - 560 \, A b^{5} c^{4} + 9552 \, B a^{2} b^{2} c^{5} + 5376 \, A a b^{3} c^{5} - 6720 \, B a^{3} c^{6} - 14592 \, A a^{2} b c^{6}}{c^{7}}\right )} x + \frac{945 \, B b^{7} c^{2} - 10500 \, B a b^{5} c^{3} - 1680 \, A b^{6} c^{3} + 37744 \, B a^{2} b^{3} c^{4} + 17920 \, A a b^{4} c^{4} - 42432 \, B a^{3} b c^{5} - 59136 \, A a^{2} b^{2} c^{5} + 49152 \, A a^{3} c^{6}}{c^{7}}\right )} + \frac{5 \,{\left (9 \, B b^{8} - 112 \, B a b^{6} c - 16 \, A b^{7} c + 480 \, B a^{2} b^{4} c^{2} + 192 \, A a b^{5} c^{2} - 768 \, B a^{3} b^{2} c^{3} - 768 \, A a^{2} b^{3} c^{3} + 256 \, B a^{4} c^{4} + 1024 \, A a^{3} b c^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{32768 \, c^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(B*x + A)*x,x, algorithm="giac")

[Out]

1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*B*c^2*x + (33*B*b*c^8 + 16
*A*c^9)/c^7)*x + (243*B*b^2*c^7 + 476*B*a*c^8 + 464*A*b*c^8)/c^7)*x + (3*B*b^3*c
^6 + 1228*B*a*b*c^7 + 592*A*b^2*c^7 + 1152*A*a*c^8)/c^7)*x - (27*B*b^4*c^5 - 216
*B*a*b^2*c^6 - 48*A*b^3*c^6 - 6608*B*a^2*c^7 - 12608*A*a*b*c^7)/c^7)*x + (63*B*b
^5*c^4 - 568*B*a*b^3*c^5 - 112*A*b^4*c^5 + 1392*B*a^2*b*c^6 + 960*A*a*b^2*c^6 +
18432*A*a^2*c^7)/c^7)*x - (315*B*b^6*c^3 - 3164*B*a*b^4*c^4 - 560*A*b^5*c^4 + 95
52*B*a^2*b^2*c^5 + 5376*A*a*b^3*c^5 - 6720*B*a^3*c^6 - 14592*A*a^2*b*c^6)/c^7)*x
+ (945*B*b^7*c^2 - 10500*B*a*b^5*c^3 - 1680*A*b^6*c^3 + 37744*B*a^2*b^3*c^4 + 1
7920*A*a*b^4*c^4 - 42432*B*a^3*b*c^5 - 59136*A*a^2*b^2*c^5 + 49152*A*a^3*c^6)/c^
7) + 5/32768*(9*B*b^8 - 112*B*a*b^6*c - 16*A*b^7*c + 480*B*a^2*b^4*c^2 + 192*A*a
*b^5*c^2 - 768*B*a^3*b^2*c^3 - 768*A*a^2*b^3*c^3 + 256*B*a^4*c^4 + 1024*A*a^3*b*
c^4)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(11/2)