### 3.1217 $$\int (b d+2 c d x)^5 (a+b x+c x^2)^{5/2} \, dx$$

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

[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

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Rubi [A]  time = 0.0508876, 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} $\frac{16}{693} d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{7/2}+\frac{8}{99} d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{2}{11} d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{7/2}$

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 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 )^{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*}

Mathematica [A]  time = 0.0940328, size = 92, normalized size = 0.94 $\frac{2}{693} d^5 (a+x (b+c x))^{7/2} \left (16 c^2 \left (8 a^2-28 a c x^2+63 c^2 x^4\right )+8 b^2 c \left (203 c x^2-22 a\right )+224 b c^2 x \left (9 c x^2-2 a\right )+616 b^3 c x+99 b^4\right )$

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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Maple [A]  time = 0.046, size = 91, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2016\,{c}^{4}{x}^{4}+4032\,b{c}^{3}{x}^{3}-896\,{x}^{2}a{c}^{3}+3248\,{x}^{2}{b}^{2}{c}^{2}-896\,xba{c}^{2}+1232\,x{b}^{3}c+256\,{a}^{2}{c}^{2}-352\,ac{b}^{2}+198\,{b}^{4} \right ){d}^{5}}{693} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{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)^(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., 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)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.03472, size = 778, normalized size = 7.94 \begin{align*} \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{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)^(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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Sympy [B]  time = 15.4882, size = 913, normalized size = 9.32 \begin{align*} \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{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)**(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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Giac [B]  time = 2.1129, size = 599, normalized size = 6.11 \begin{align*} \frac{2}{693} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left ({\left ({\left ({\left ({\left (56 \,{\left ({\left (18 \,{\left (c^{7} d^{5} x + 5 \, b c^{6} d^{5}\right )} x + \frac{191 \, b^{2} c^{15} d^{5} + 46 \, a c^{16} d^{5}}{c^{10}}\right )} x + \frac{8 \,{\left (28 \, b^{3} c^{14} d^{5} + 23 \, a b c^{15} d^{5}\right )}}{c^{10}}\right )} x + \frac{8835 \, b^{4} c^{13} d^{5} + 17128 \, a b^{2} c^{14} d^{5} + 1808 \, a^{2} c^{15} d^{5}}{c^{10}}\right )} x + \frac{3769 \, b^{5} c^{12} d^{5} + 15320 \, a b^{3} c^{13} d^{5} + 5424 \, a^{2} b c^{14} d^{5}}{c^{10}}\right )} x + \frac{913 \, b^{6} c^{11} d^{5} + 7889 \, a b^{4} c^{12} d^{5} + 6744 \, a^{2} b^{2} c^{13} d^{5} + 48 \, a^{3} c^{14} d^{5}}{c^{10}}\right )} x + \frac{99 \, b^{7} c^{10} d^{5} + 2266 \, a b^{5} c^{11} d^{5} + 4448 \, a^{2} b^{3} c^{12} d^{5} + 96 \, a^{3} b c^{13} d^{5}}{c^{10}}\right )} x + \frac{297 \, a b^{6} c^{10} d^{5} + 1617 \, a^{2} b^{4} c^{11} d^{5} + 136 \, a^{3} b^{2} c^{12} d^{5} - 64 \, a^{4} c^{13} d^{5}}{c^{10}}\right )} x + \frac{297 \, a^{2} b^{5} c^{10} d^{5} + 88 \, a^{3} b^{3} c^{11} d^{5} - 64 \, a^{4} b c^{12} d^{5}}{c^{10}}\right )} x + \frac{99 \, a^{3} b^{4} c^{10} d^{5} - 176 \, a^{4} b^{2} c^{11} d^{5} + 128 \, a^{5} c^{12} d^{5}}{c^{10}}\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)^(5/2),x, algorithm="giac")

[Out]

2/693*sqrt(c*x^2 + b*x + a)*(((((((56*((18*(c^7*d^5*x + 5*b*c^6*d^5)*x + (191*b^2*c^15*d^5 + 46*a*c^16*d^5)/c^
10)*x + 8*(28*b^3*c^14*d^5 + 23*a*b*c^15*d^5)/c^10)*x + (8835*b^4*c^13*d^5 + 17128*a*b^2*c^14*d^5 + 1808*a^2*c
^15*d^5)/c^10)*x + (3769*b^5*c^12*d^5 + 15320*a*b^3*c^13*d^5 + 5424*a^2*b*c^14*d^5)/c^10)*x + (913*b^6*c^11*d^
5 + 7889*a*b^4*c^12*d^5 + 6744*a^2*b^2*c^13*d^5 + 48*a^3*c^14*d^5)/c^10)*x + (99*b^7*c^10*d^5 + 2266*a*b^5*c^1
1*d^5 + 4448*a^2*b^3*c^12*d^5 + 96*a^3*b*c^13*d^5)/c^10)*x + (297*a*b^6*c^10*d^5 + 1617*a^2*b^4*c^11*d^5 + 136
*a^3*b^2*c^12*d^5 - 64*a^4*c^13*d^5)/c^10)*x + (297*a^2*b^5*c^10*d^5 + 88*a^3*b^3*c^11*d^5 - 64*a^4*b*c^12*d^5
)/c^10)*x + (99*a^3*b^4*c^10*d^5 - 176*a^4*b^2*c^11*d^5 + 128*a^5*c^12*d^5)/c^10)