3.1202 \(\int (b d+2 c d x)^5 (a+b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=98 \[ \frac{16}{315} d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}+\frac{8}{63} d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2} \]

[Out]

(16*(b^2 - 4*a*c)^2*d^5*(a + b*x + c*x^2)^(5/2))/315 + (8*(b^2 - 4*a*c)*d^5*(b + 2*c*x)^2*(a + b*x + c*x^2)^(5
/2))/63 + (2*d^5*(b + 2*c*x)^4*(a + b*x + c*x^2)^(5/2))/9

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Rubi [A]  time = 0.0512991, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {692, 629} \[ \frac{16}{315} d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{5/2}+\frac{8}{63} d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(b^2 - 4*a*c)^2*d^5*(a + b*x + c*x^2)^(5/2))/315 + (8*(b^2 - 4*a*c)*d^5*(b + 2*c*x)^2*(a + b*x + c*x^2)^(5
/2))/63 + (2*d^5*(b + 2*c*x)^4*(a + b*x + c*x^2)^(5/2))/9

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*d*(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1))/(b*(m + 2*p + 1)), x] + Dist[(d^2*(m - 1)*(b^2 - 4*a*c))/(b^2*(m + 2*p + 1)), In
t[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[
2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] &
& RationalQ[p]) || OddQ[m])

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2}+\frac{1}{9} \left (4 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx\\ &=\frac{8}{63} \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2}+\frac{1}{63} \left (8 \left (b^2-4 a c\right )^2 d^4\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^{3/2} \, dx\\ &=\frac{16}{315} \left (b^2-4 a c\right )^2 d^5 \left (a+b x+c x^2\right )^{5/2}+\frac{8}{63} \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac{2}{9} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0773778, size = 92, normalized size = 0.94 \[ \frac{2}{315} d^5 (a+x (b+c x))^{5/2} \left (16 c^2 \left (8 a^2-20 a c x^2+35 c^2 x^4\right )+8 b^2 c \left (115 c x^2-18 a\right )+160 b c^2 x \left (7 c x^2-2 a\right )+360 b^3 c x+63 b^4\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^5*(a + x*(b + c*x))^(5/2)*(63*b^4 + 360*b^3*c*x + 160*b*c^2*x*(-2*a + 7*c*x^2) + 8*b^2*c*(-18*a + 115*c*x
^2) + 16*c^2*(8*a^2 - 20*a*c*x^2 + 35*c^2*x^4)))/315

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Maple [A]  time = 0.045, size = 91, normalized size = 0.9 \begin{align*}{\frac{ \left ( 1120\,{c}^{4}{x}^{4}+2240\,b{c}^{3}{x}^{3}-640\,{x}^{2}a{c}^{3}+1840\,{x}^{2}{b}^{2}{c}^{2}-640\,xba{c}^{2}+720\,x{b}^{3}c+256\,{a}^{2}{c}^{2}-288\,ac{b}^{2}+126\,{b}^{4} \right ){d}^{5}}{315} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(3/2),x)

[Out]

2/315*(c*x^2+b*x+a)^(5/2)*(560*c^4*x^4+1120*b*c^3*x^3-320*a*c^3*x^2+920*b^2*c^2*x^2-320*a*b*c^2*x+360*b^3*c*x+
128*a^2*c^2-144*a*b^2*c+63*b^4)*d^5

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.93849, size = 564, normalized size = 5.76 \begin{align*} \frac{2}{315} \,{\left (560 \, c^{6} d^{5} x^{8} + 2240 \, b c^{5} d^{5} x^{7} + 40 \,{\left (93 \, b^{2} c^{4} + 20 \, a c^{5}\right )} d^{5} x^{6} + 40 \,{\left (83 \, b^{3} c^{3} + 60 \, a b c^{4}\right )} d^{5} x^{5} +{\left (1703 \, b^{4} c^{2} + 2976 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} d^{5} x^{4} + 2 \,{\left (243 \, b^{5} c + 976 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} d^{5} x^{3} +{\left (63 \, b^{6} + 702 \, a b^{4} c + 120 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{5} x^{2} + 2 \,{\left (63 \, a b^{5} + 36 \, a^{2} b^{3} c - 32 \, a^{3} b c^{2}\right )} d^{5} x +{\left (63 \, a^{2} b^{4} - 144 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} d^{5}\right )} \sqrt{c x^{2} + b x + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

2/315*(560*c^6*d^5*x^8 + 2240*b*c^5*d^5*x^7 + 40*(93*b^2*c^4 + 20*a*c^5)*d^5*x^6 + 40*(83*b^3*c^3 + 60*a*b*c^4
)*d^5*x^5 + (1703*b^4*c^2 + 2976*a*b^2*c^3 + 48*a^2*c^4)*d^5*x^4 + 2*(243*b^5*c + 976*a*b^3*c^2 + 48*a^2*b*c^3
)*d^5*x^3 + (63*b^6 + 702*a*b^4*c + 120*a^2*b^2*c^2 - 64*a^3*c^3)*d^5*x^2 + 2*(63*a*b^5 + 36*a^2*b^3*c - 32*a^
3*b*c^2)*d^5*x + (63*a^2*b^4 - 144*a^3*b^2*c + 128*a^4*c^2)*d^5)*sqrt(c*x^2 + b*x + a)

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Sympy [B]  time = 4.43183, size = 656, normalized size = 6.69 \begin{align*} \frac{256 a^{4} c^{2} d^{5} \sqrt{a + b x + c x^{2}}}{315} - \frac{32 a^{3} b^{2} c d^{5} \sqrt{a + b x + c x^{2}}}{35} - \frac{128 a^{3} b c^{2} d^{5} x \sqrt{a + b x + c x^{2}}}{315} - \frac{128 a^{3} c^{3} d^{5} x^{2} \sqrt{a + b x + c x^{2}}}{315} + \frac{2 a^{2} b^{4} d^{5} \sqrt{a + b x + c x^{2}}}{5} + \frac{16 a^{2} b^{3} c d^{5} x \sqrt{a + b x + c x^{2}}}{35} + \frac{16 a^{2} b^{2} c^{2} d^{5} x^{2} \sqrt{a + b x + c x^{2}}}{21} + \frac{64 a^{2} b c^{3} d^{5} x^{3} \sqrt{a + b x + c x^{2}}}{105} + \frac{32 a^{2} c^{4} d^{5} x^{4} \sqrt{a + b x + c x^{2}}}{105} + \frac{4 a b^{5} d^{5} x \sqrt{a + b x + c x^{2}}}{5} + \frac{156 a b^{4} c d^{5} x^{2} \sqrt{a + b x + c x^{2}}}{35} + \frac{3904 a b^{3} c^{2} d^{5} x^{3} \sqrt{a + b x + c x^{2}}}{315} + \frac{1984 a b^{2} c^{3} d^{5} x^{4} \sqrt{a + b x + c x^{2}}}{105} + \frac{320 a b c^{4} d^{5} x^{5} \sqrt{a + b x + c x^{2}}}{21} + \frac{320 a c^{5} d^{5} x^{6} \sqrt{a + b x + c x^{2}}}{63} + \frac{2 b^{6} d^{5} x^{2} \sqrt{a + b x + c x^{2}}}{5} + \frac{108 b^{5} c d^{5} x^{3} \sqrt{a + b x + c x^{2}}}{35} + \frac{3406 b^{4} c^{2} d^{5} x^{4} \sqrt{a + b x + c x^{2}}}{315} + \frac{1328 b^{3} c^{3} d^{5} x^{5} \sqrt{a + b x + c x^{2}}}{63} + \frac{496 b^{2} c^{4} d^{5} x^{6} \sqrt{a + b x + c x^{2}}}{21} + \frac{128 b c^{5} d^{5} x^{7} \sqrt{a + b x + c x^{2}}}{9} + \frac{32 c^{6} d^{5} x^{8} \sqrt{a + b x + c x^{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**(3/2),x)

[Out]

256*a**4*c**2*d**5*sqrt(a + b*x + c*x**2)/315 - 32*a**3*b**2*c*d**5*sqrt(a + b*x + c*x**2)/35 - 128*a**3*b*c**
2*d**5*x*sqrt(a + b*x + c*x**2)/315 - 128*a**3*c**3*d**5*x**2*sqrt(a + b*x + c*x**2)/315 + 2*a**2*b**4*d**5*sq
rt(a + b*x + c*x**2)/5 + 16*a**2*b**3*c*d**5*x*sqrt(a + b*x + c*x**2)/35 + 16*a**2*b**2*c**2*d**5*x**2*sqrt(a
+ b*x + c*x**2)/21 + 64*a**2*b*c**3*d**5*x**3*sqrt(a + b*x + c*x**2)/105 + 32*a**2*c**4*d**5*x**4*sqrt(a + b*x
+ c*x**2)/105 + 4*a*b**5*d**5*x*sqrt(a + b*x + c*x**2)/5 + 156*a*b**4*c*d**5*x**2*sqrt(a + b*x + c*x**2)/35 +
3904*a*b**3*c**2*d**5*x**3*sqrt(a + b*x + c*x**2)/315 + 1984*a*b**2*c**3*d**5*x**4*sqrt(a + b*x + c*x**2)/105
+ 320*a*b*c**4*d**5*x**5*sqrt(a + b*x + c*x**2)/21 + 320*a*c**5*d**5*x**6*sqrt(a + b*x + c*x**2)/63 + 2*b**6*
d**5*x**2*sqrt(a + b*x + c*x**2)/5 + 108*b**5*c*d**5*x**3*sqrt(a + b*x + c*x**2)/35 + 3406*b**4*c**2*d**5*x**4
*sqrt(a + b*x + c*x**2)/315 + 1328*b**3*c**3*d**5*x**5*sqrt(a + b*x + c*x**2)/63 + 496*b**2*c**4*d**5*x**6*sqr
t(a + b*x + c*x**2)/21 + 128*b*c**5*d**5*x**7*sqrt(a + b*x + c*x**2)/9 + 32*c**6*d**5*x**8*sqrt(a + b*x + c*x*
*2)/9

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Giac [B]  time = 1.19533, size = 441, normalized size = 4.5 \begin{align*} \frac{2}{315} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left ({\left ({\left (40 \,{\left ({\left (14 \,{\left (c^{6} d^{5} x + 4 \, b c^{5} d^{5}\right )} x + \frac{93 \, b^{2} c^{12} d^{5} + 20 \, a c^{13} d^{5}}{c^{8}}\right )} x + \frac{83 \, b^{3} c^{11} d^{5} + 60 \, a b c^{12} d^{5}}{c^{8}}\right )} x + \frac{1703 \, b^{4} c^{10} d^{5} + 2976 \, a b^{2} c^{11} d^{5} + 48 \, a^{2} c^{12} d^{5}}{c^{8}}\right )} x + \frac{2 \,{\left (243 \, b^{5} c^{9} d^{5} + 976 \, a b^{3} c^{10} d^{5} + 48 \, a^{2} b c^{11} d^{5}\right )}}{c^{8}}\right )} x + \frac{63 \, b^{6} c^{8} d^{5} + 702 \, a b^{4} c^{9} d^{5} + 120 \, a^{2} b^{2} c^{10} d^{5} - 64 \, a^{3} c^{11} d^{5}}{c^{8}}\right )} x + \frac{2 \,{\left (63 \, a b^{5} c^{8} d^{5} + 36 \, a^{2} b^{3} c^{9} d^{5} - 32 \, a^{3} b c^{10} d^{5}\right )}}{c^{8}}\right )} x + \frac{63 \, a^{2} b^{4} c^{8} d^{5} - 144 \, a^{3} b^{2} c^{9} d^{5} + 128 \, a^{4} c^{10} d^{5}}{c^{8}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

2/315*sqrt(c*x^2 + b*x + a)*(((((40*((14*(c^6*d^5*x + 4*b*c^5*d^5)*x + (93*b^2*c^12*d^5 + 20*a*c^13*d^5)/c^8)*
x + (83*b^3*c^11*d^5 + 60*a*b*c^12*d^5)/c^8)*x + (1703*b^4*c^10*d^5 + 2976*a*b^2*c^11*d^5 + 48*a^2*c^12*d^5)/c
^8)*x + 2*(243*b^5*c^9*d^5 + 976*a*b^3*c^10*d^5 + 48*a^2*b*c^11*d^5)/c^8)*x + (63*b^6*c^8*d^5 + 702*a*b^4*c^9*
d^5 + 120*a^2*b^2*c^10*d^5 - 64*a^3*c^11*d^5)/c^8)*x + 2*(63*a*b^5*c^8*d^5 + 36*a^2*b^3*c^9*d^5 - 32*a^3*b*c^1
0*d^5)/c^8)*x + (63*a^2*b^4*c^8*d^5 - 144*a^3*b^2*c^9*d^5 + 128*a^4*c^10*d^5)/c^8)