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

Optimal. Leaf size=98 \[ \frac {16}{693} d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{7/2}+\frac {8}{99} d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2} \]

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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}{693} d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{7/2}+\frac {8}{99} d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(b^2 - 4*a*c)^2*d^5*(a + b*x + c*x^2)^(7/2))/693 + (8*(b^2 - 4*a*c)*d^5*(b + 2*c*x)^2*(a + b*x + c*x^2)^(7
/2))/99 + (2*d^5*(b + 2*c*x)^4*(a + b*x + c*x^2)^(7/2))/11

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 )^{5/2} \, dx &=\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2}+\frac {1}{11} \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 )^{5/2} \, dx\\ &=\frac {8}{99} \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2}+\frac {1}{99} \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 )^{5/2} \, dx\\ &=\frac {16}{693} \left (b^2-4 a c\right )^2 d^5 \left (a+b x+c x^2\right )^{7/2}+\frac {8}{99} \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 92, normalized size = 0.94 \begin {gather*} \frac {2}{693} d^5 (a+x (b+c x))^{7/2} \left (16 c^2 \left (8 a^2-28 a c x^2+63 c^2 x^4\right )+8 b^2 c \left (203 c x^2-22 a\right )+224 b c^2 x \left (9 c x^2-2 a\right )+99 b^4+616 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)^(5/2),x]

[Out]

(2*d^5*(a + x*(b + c*x))^(7/2)*(99*b^4 + 616*b^3*c*x + 224*b*c^2*x*(-2*a + 9*c*x^2) + 8*b^2*c*(-22*a + 203*c*x
^2) + 16*c^2*(8*a^2 - 28*a*c*x^2 + 63*c^2*x^4)))/693

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IntegrateAlgebraic [B]  time = 1.60, size = 426, normalized size = 4.35 \begin {gather*} \frac {2}{693} \sqrt {a+b x+c x^2} \left (128 a^5 c^2 d^5-176 a^4 b^2 c d^5-64 a^4 b c^2 d^5 x-64 a^4 c^3 d^5 x^2+99 a^3 b^4 d^5+88 a^3 b^3 c d^5 x+136 a^3 b^2 c^2 d^5 x^2+96 a^3 b c^3 d^5 x^3+48 a^3 c^4 d^5 x^4+297 a^2 b^5 d^5 x+1617 a^2 b^4 c d^5 x^2+4448 a^2 b^3 c^2 d^5 x^3+6744 a^2 b^2 c^3 d^5 x^4+5424 a^2 b c^4 d^5 x^5+1808 a^2 c^5 d^5 x^6+297 a b^6 d^5 x^2+2266 a b^5 c d^5 x^3+7889 a b^4 c^2 d^5 x^4+15320 a b^3 c^3 d^5 x^5+17128 a b^2 c^4 d^5 x^6+10304 a b c^5 d^5 x^7+2576 a c^6 d^5 x^8+99 b^7 d^5 x^3+913 b^6 c d^5 x^4+3769 b^5 c^2 d^5 x^5+8835 b^4 c^3 d^5 x^6+12544 b^3 c^4 d^5 x^7+10696 b^2 c^5 d^5 x^8+5040 b c^6 d^5 x^9+1008 c^7 d^5 x^{10}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a + b*x + c*x^2]*(99*a^3*b^4*d^5 - 176*a^4*b^2*c*d^5 + 128*a^5*c^2*d^5 + 297*a^2*b^5*d^5*x + 88*a^3*b^
3*c*d^5*x - 64*a^4*b*c^2*d^5*x + 297*a*b^6*d^5*x^2 + 1617*a^2*b^4*c*d^5*x^2 + 136*a^3*b^2*c^2*d^5*x^2 - 64*a^4
*c^3*d^5*x^2 + 99*b^7*d^5*x^3 + 2266*a*b^5*c*d^5*x^3 + 4448*a^2*b^3*c^2*d^5*x^3 + 96*a^3*b*c^3*d^5*x^3 + 913*b
^6*c*d^5*x^4 + 7889*a*b^4*c^2*d^5*x^4 + 6744*a^2*b^2*c^3*d^5*x^4 + 48*a^3*c^4*d^5*x^4 + 3769*b^5*c^2*d^5*x^5 +
15320*a*b^3*c^3*d^5*x^5 + 5424*a^2*b*c^4*d^5*x^5 + 8835*b^4*c^3*d^5*x^6 + 17128*a*b^2*c^4*d^5*x^6 + 1808*a^2*
c^5*d^5*x^6 + 12544*b^3*c^4*d^5*x^7 + 10304*a*b*c^5*d^5*x^7 + 10696*b^2*c^5*d^5*x^8 + 2576*a*c^6*d^5*x^8 + 504
0*b*c^6*d^5*x^9 + 1008*c^7*d^5*x^10))/693

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fricas [B]  time = 0.49, size = 338, normalized size = 3.45 \begin {gather*} \frac {2}{693} \, {\left (1008 \, c^{7} d^{5} x^{10} + 5040 \, b c^{6} d^{5} x^{9} + 56 \, {\left (191 \, b^{2} c^{5} + 46 \, a c^{6}\right )} d^{5} x^{8} + 448 \, {\left (28 \, b^{3} c^{4} + 23 \, a b c^{5}\right )} d^{5} x^{7} + {\left (8835 \, b^{4} c^{3} + 17128 \, a b^{2} c^{4} + 1808 \, a^{2} c^{5}\right )} d^{5} x^{6} + {\left (3769 \, b^{5} c^{2} + 15320 \, a b^{3} c^{3} + 5424 \, a^{2} b c^{4}\right )} d^{5} x^{5} + {\left (913 \, b^{6} c + 7889 \, a b^{4} c^{2} + 6744 \, a^{2} b^{2} c^{3} + 48 \, a^{3} c^{4}\right )} d^{5} x^{4} + {\left (99 \, b^{7} + 2266 \, a b^{5} c + 4448 \, a^{2} b^{3} c^{2} + 96 \, a^{3} b c^{3}\right )} d^{5} x^{3} + {\left (297 \, a b^{6} + 1617 \, a^{2} b^{4} c + 136 \, a^{3} b^{2} c^{2} - 64 \, a^{4} c^{3}\right )} d^{5} x^{2} + {\left (297 \, a^{2} b^{5} + 88 \, a^{3} b^{3} c - 64 \, a^{4} b c^{2}\right )} d^{5} x + {\left (99 \, a^{3} b^{4} - 176 \, a^{4} b^{2} c + 128 \, a^{5} 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)^(5/2),x, algorithm="fricas")

[Out]

2/693*(1008*c^7*d^5*x^10 + 5040*b*c^6*d^5*x^9 + 56*(191*b^2*c^5 + 46*a*c^6)*d^5*x^8 + 448*(28*b^3*c^4 + 23*a*b
*c^5)*d^5*x^7 + (8835*b^4*c^3 + 17128*a*b^2*c^4 + 1808*a^2*c^5)*d^5*x^6 + (3769*b^5*c^2 + 15320*a*b^3*c^3 + 54
24*a^2*b*c^4)*d^5*x^5 + (913*b^6*c + 7889*a*b^4*c^2 + 6744*a^2*b^2*c^3 + 48*a^3*c^4)*d^5*x^4 + (99*b^7 + 2266*
a*b^5*c + 4448*a^2*b^3*c^2 + 96*a^3*b*c^3)*d^5*x^3 + (297*a*b^6 + 1617*a^2*b^4*c + 136*a^3*b^2*c^2 - 64*a^4*c^
3)*d^5*x^2 + (297*a^2*b^5 + 88*a^3*b^3*c - 64*a^4*b*c^2)*d^5*x + (99*a^3*b^4 - 176*a^4*b^2*c + 128*a^5*c^2)*d^
5)*sqrt(c*x^2 + b*x + a)

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giac [A]  time = 0.19, size = 128, normalized size = 1.31 \begin {gather*} \frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} b^{4} d^{5} + \frac {16}{9} \, {\left (c x^{2} + b x + a\right )}^{\frac {9}{2}} b^{2} c d^{5} - \frac {16}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} a b^{2} c d^{5} + \frac {32}{11} \, {\left (c x^{2} + b x + a\right )}^{\frac {11}{2}} c^{2} d^{5} - \frac {64}{9} \, {\left (c x^{2} + b x + a\right )}^{\frac {9}{2}} a c^{2} d^{5} + \frac {32}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{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)^(5/2),x, algorithm="giac")

[Out]

2/7*(c*x^2 + b*x + a)^(7/2)*b^4*d^5 + 16/9*(c*x^2 + b*x + a)^(9/2)*b^2*c*d^5 - 16/7*(c*x^2 + b*x + a)^(7/2)*a*
b^2*c*d^5 + 32/11*(c*x^2 + b*x + a)^(11/2)*c^2*d^5 - 64/9*(c*x^2 + b*x + a)^(9/2)*a*c^2*d^5 + 32/7*(c*x^2 + b*
x + a)^(7/2)*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 {7}{2}} \left (1008 c^{4} x^{4}+2016 b \,c^{3} x^{3}-448 a \,c^{3} x^{2}+1624 x^{2} b^{2} c^{2}-448 x b a \,c^{2}+616 x \,b^{3} c +128 a^{2} c^{2}-176 a \,b^{2} c +99 b^{4}\right ) d^{5}}{693} \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)^(5/2),x)

[Out]

2/693*(c*x^2+b*x+a)^(7/2)*(1008*c^4*x^4+2016*b*c^3*x^3-448*a*c^3*x^2+1624*b^2*c^2*x^2-448*a*b*c^2*x+616*b^3*c*
x+128*a^2*c^2-176*a*b^2*c+99*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)^(5/2),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 = 1.41, size = 128, normalized size = 1.31 \begin {gather*} \frac {2\,b^4\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7}+\frac {32\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{11/2}}{11}-\frac {64\,a\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{9/2}}{9}+\frac {16\,b^2\,c\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{9/2}}{9}+\frac {32\,a^2\,c^2\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7}-\frac {16\,a\,b^2\,c\,d^5\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7} \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)^(5/2),x)

[Out]

(2*b^4*d^5*(a + b*x + c*x^2)^(7/2))/7 + (32*c^2*d^5*(a + b*x + c*x^2)^(11/2))/11 - (64*a*c^2*d^5*(a + b*x + c*
x^2)^(9/2))/9 + (16*b^2*c*d^5*(a + b*x + c*x^2)^(9/2))/9 + (32*a^2*c^2*d^5*(a + b*x + c*x^2)^(7/2))/7 - (16*a*
b^2*c*d^5*(a + b*x + c*x^2)^(7/2))/7

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sympy [B]  time = 10.08, size = 913, normalized size = 9.32 \begin {gather*} \frac {256 a^{5} c^{2} d^{5} \sqrt {a + b x + c x^{2}}}{693} - \frac {32 a^{4} b^{2} c d^{5} \sqrt {a + b x + c x^{2}}}{63} - \frac {128 a^{4} b c^{2} d^{5} x \sqrt {a + b x + c x^{2}}}{693} - \frac {128 a^{4} c^{3} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{693} + \frac {2 a^{3} b^{4} d^{5} \sqrt {a + b x + c x^{2}}}{7} + \frac {16 a^{3} b^{3} c d^{5} x \sqrt {a + b x + c x^{2}}}{63} + \frac {272 a^{3} b^{2} c^{2} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{693} + \frac {64 a^{3} b c^{3} d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{231} + \frac {32 a^{3} c^{4} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{231} + \frac {6 a^{2} b^{5} d^{5} x \sqrt {a + b x + c x^{2}}}{7} + \frac {14 a^{2} b^{4} c d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{3} + \frac {8896 a^{2} b^{3} c^{2} d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{693} + \frac {4496 a^{2} b^{2} c^{3} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{231} + \frac {3616 a^{2} b c^{4} d^{5} x^{5} \sqrt {a + b x + c x^{2}}}{231} + \frac {3616 a^{2} c^{5} d^{5} x^{6} \sqrt {a + b x + c x^{2}}}{693} + \frac {6 a b^{6} d^{5} x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {412 a b^{5} c d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{63} + \frac {2254 a b^{4} c^{2} d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{99} + \frac {30640 a b^{3} c^{3} d^{5} x^{5} \sqrt {a + b x + c x^{2}}}{693} + \frac {34256 a b^{2} c^{4} d^{5} x^{6} \sqrt {a + b x + c x^{2}}}{693} + \frac {2944 a b c^{5} d^{5} x^{7} \sqrt {a + b x + c x^{2}}}{99} + \frac {736 a c^{6} d^{5} x^{8} \sqrt {a + b x + c x^{2}}}{99} + \frac {2 b^{7} d^{5} x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {166 b^{6} c d^{5} x^{4} \sqrt {a + b x + c x^{2}}}{63} + \frac {7538 b^{5} c^{2} d^{5} x^{5} \sqrt {a + b x + c x^{2}}}{693} + \frac {5890 b^{4} c^{3} d^{5} x^{6} \sqrt {a + b x + c x^{2}}}{231} + \frac {3584 b^{3} c^{4} d^{5} x^{7} \sqrt {a + b x + c x^{2}}}{99} + \frac {3056 b^{2} c^{5} d^{5} x^{8} \sqrt {a + b x + c x^{2}}}{99} + \frac {160 b c^{6} d^{5} x^{9} \sqrt {a + b x + c x^{2}}}{11} + \frac {32 c^{7} d^{5} x^{10} \sqrt {a + b x + c x^{2}}}{11} \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)**(5/2),x)

[Out]

256*a**5*c**2*d**5*sqrt(a + b*x + c*x**2)/693 - 32*a**4*b**2*c*d**5*sqrt(a + b*x + c*x**2)/63 - 128*a**4*b*c**
2*d**5*x*sqrt(a + b*x + c*x**2)/693 - 128*a**4*c**3*d**5*x**2*sqrt(a + b*x + c*x**2)/693 + 2*a**3*b**4*d**5*sq
rt(a + b*x + c*x**2)/7 + 16*a**3*b**3*c*d**5*x*sqrt(a + b*x + c*x**2)/63 + 272*a**3*b**2*c**2*d**5*x**2*sqrt(a
+ b*x + c*x**2)/693 + 64*a**3*b*c**3*d**5*x**3*sqrt(a + b*x + c*x**2)/231 + 32*a**3*c**4*d**5*x**4*sqrt(a + b
*x + c*x**2)/231 + 6*a**2*b**5*d**5*x*sqrt(a + b*x + c*x**2)/7 + 14*a**2*b**4*c*d**5*x**2*sqrt(a + b*x + c*x**
2)/3 + 8896*a**2*b**3*c**2*d**5*x**3*sqrt(a + b*x + c*x**2)/693 + 4496*a**2*b**2*c**3*d**5*x**4*sqrt(a + b*x +
c*x**2)/231 + 3616*a**2*b*c**4*d**5*x**5*sqrt(a + b*x + c*x**2)/231 + 3616*a**2*c**5*d**5*x**6*sqrt(a + b*x +
c*x**2)/693 + 6*a*b**6*d**5*x**2*sqrt(a + b*x + c*x**2)/7 + 412*a*b**5*c*d**5*x**3*sqrt(a + b*x + c*x**2)/63
+ 2254*a*b**4*c**2*d**5*x**4*sqrt(a + b*x + c*x**2)/99 + 30640*a*b**3*c**3*d**5*x**5*sqrt(a + b*x + c*x**2)/69
3 + 34256*a*b**2*c**4*d**5*x**6*sqrt(a + b*x + c*x**2)/693 + 2944*a*b*c**5*d**5*x**7*sqrt(a + b*x + c*x**2)/99
+ 736*a*c**6*d**5*x**8*sqrt(a + b*x + c*x**2)/99 + 2*b**7*d**5*x**3*sqrt(a + b*x + c*x**2)/7 + 166*b**6*c*d**
5*x**4*sqrt(a + b*x + c*x**2)/63 + 7538*b**5*c**2*d**5*x**5*sqrt(a + b*x + c*x**2)/693 + 5890*b**4*c**3*d**5*x
**6*sqrt(a + b*x + c*x**2)/231 + 3584*b**3*c**4*d**5*x**7*sqrt(a + b*x + c*x**2)/99 + 3056*b**2*c**5*d**5*x**8
*sqrt(a + b*x + c*x**2)/99 + 160*b*c**6*d**5*x**9*sqrt(a + b*x + c*x**2)/11 + 32*c**7*d**5*x**10*sqrt(a + b*x
+ c*x**2)/11

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