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

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

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Rubi [A]  time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 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]

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])

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*}

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Mathematica [A]  time = 0.08, size = 92, normalized size = 0.94 \begin {gather*} \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 )+63 b^4+360 b^3 c x\right ) \end {gather*}

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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IntegrateAlgebraic [B]  time = 1.04, size = 305, normalized size = 3.11 \begin {gather*} \frac {2}{315} \sqrt {a+b x+c x^2} \left (128 a^4 c^2 d^5-144 a^3 b^2 c d^5-64 a^3 b c^2 d^5 x-64 a^3 c^3 d^5 x^2+63 a^2 b^4 d^5+72 a^2 b^3 c d^5 x+120 a^2 b^2 c^2 d^5 x^2+96 a^2 b c^3 d^5 x^3+48 a^2 c^4 d^5 x^4+126 a b^5 d^5 x+702 a b^4 c d^5 x^2+1952 a b^3 c^2 d^5 x^3+2976 a b^2 c^3 d^5 x^4+2400 a b c^4 d^5 x^5+800 a c^5 d^5 x^6+63 b^6 d^5 x^2+486 b^5 c d^5 x^3+1703 b^4 c^2 d^5 x^4+3320 b^3 c^3 d^5 x^5+3720 b^2 c^4 d^5 x^6+2240 b c^5 d^5 x^7+560 c^6 d^5 x^8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*x + c*x^2]*(63*a^2*b^4*d^5 - 144*a^3*b^2*c*d^5 + 128*a^4*c^2*d^5 + 126*a*b^5*d^5*x + 72*a^2*b^3*
c*d^5*x - 64*a^3*b*c^2*d^5*x + 63*b^6*d^5*x^2 + 702*a*b^4*c*d^5*x^2 + 120*a^2*b^2*c^2*d^5*x^2 - 64*a^3*c^3*d^5
*x^2 + 486*b^5*c*d^5*x^3 + 1952*a*b^3*c^2*d^5*x^3 + 96*a^2*b*c^3*d^5*x^3 + 1703*b^4*c^2*d^5*x^4 + 2976*a*b^2*c
^3*d^5*x^4 + 48*a^2*c^4*d^5*x^4 + 3320*b^3*c^3*d^5*x^5 + 2400*a*b*c^4*d^5*x^5 + 3720*b^2*c^4*d^5*x^6 + 800*a*c
^5*d^5*x^6 + 2240*b*c^5*d^5*x^7 + 560*c^6*d^5*x^8))/315

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fricas [B]  time = 0.45, size = 251, normalized size = 2.56 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.24, size = 128, normalized size = 1.31 \begin {gather*} \frac {2}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} b^{4} d^{5} + \frac {16}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} b^{2} c d^{5} - \frac {16}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} a b^{2} c d^{5} + \frac {32}{9} \, {\left (c x^{2} + b x + a\right )}^{\frac {9}{2}} c^{2} d^{5} - \frac {64}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} a c^{2} d^{5} + \frac {32}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} a^{2} c^{2} d^{5} \end {gather*}

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]

2/5*(c*x^2 + b*x + a)^(5/2)*b^4*d^5 + 16/7*(c*x^2 + b*x + a)^(7/2)*b^2*c*d^5 - 16/5*(c*x^2 + b*x + a)^(5/2)*a*
b^2*c*d^5 + 32/9*(c*x^2 + b*x + a)^(9/2)*c^2*d^5 - 64/7*(c*x^2 + b*x + a)^(7/2)*a*c^2*d^5 + 32/5*(c*x^2 + b*x
+ a)^(5/2)*a^2*c^2*d^5

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maple [A]  time = 0.05, size = 91, normalized size = 0.93 \begin {gather*} \frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (560 c^{4} x^{4}+1120 b \,c^{3} x^{3}-320 x^{2} a \,c^{3}+920 x^{2} b^{2} c^{2}-320 x b a \,c^{2}+360 x \,b^{3} c +128 a^{2} c^{2}-144 a \,b^{2} c +63 b^{4}\right ) d^{5}}{315} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive, negative or zero?

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mupad [B]  time = 0.97, size = 241, normalized size = 2.46 \begin {gather*} \sqrt {c\,x^2+b\,x+a}\,\left (\frac {2\,a^2\,d^5\,\left (128\,a^2\,c^2-144\,a\,b^2\,c+63\,b^4\right )}{315}+\frac {32\,c^6\,d^5\,x^8}{9}+\frac {2\,d^5\,x^2\,\left (-64\,a^3\,c^3+120\,a^2\,b^2\,c^2+702\,a\,b^4\,c+63\,b^6\right )}{315}+\frac {2\,c^2\,d^5\,x^4\,\left (48\,a^2\,c^2+2976\,a\,b^2\,c+1703\,b^4\right )}{315}+\frac {128\,b\,c^5\,d^5\,x^7}{9}+\frac {16\,c^4\,d^5\,x^6\,\left (93\,b^2+20\,a\,c\right )}{63}+\frac {4\,b\,c\,d^5\,x^3\,\left (48\,a^2\,c^2+976\,a\,b^2\,c+243\,b^4\right )}{315}+\frac {4\,a\,b\,d^5\,x\,\left (-32\,a^2\,c^2+36\,a\,b^2\,c+63\,b^4\right )}{315}+\frac {16\,b\,c^3\,d^5\,x^5\,\left (83\,b^2+60\,a\,c\right )}{63}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a + b*x + c*x^2)^(1/2)*((2*a^2*d^5*(63*b^4 + 128*a^2*c^2 - 144*a*b^2*c))/315 + (32*c^6*d^5*x^8)/9 + (2*d^5*x^
2*(63*b^6 - 64*a^3*c^3 + 120*a^2*b^2*c^2 + 702*a*b^4*c))/315 + (2*c^2*d^5*x^4*(1703*b^4 + 48*a^2*c^2 + 2976*a*
b^2*c))/315 + (128*b*c^5*d^5*x^7)/9 + (16*c^4*d^5*x^6*(20*a*c + 93*b^2))/63 + (4*b*c*d^5*x^3*(243*b^4 + 48*a^2
*c^2 + 976*a*b^2*c))/315 + (4*a*b*d^5*x*(63*b^4 - 32*a^2*c^2 + 36*a*b^2*c))/315 + (16*b*c^3*d^5*x^5*(60*a*c +
83*b^2))/63)

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sympy [B]  time = 3.49, size = 656, normalized size = 6.69 \begin {gather*} \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 {gather*}

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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