### 3.1231 $$\int \frac{(a+b x+c x^2)^{5/2}}{(b d+2 c d x)^{10}} \, dx$$

Optimal. Leaf size=79 $\frac{4 \left (a+b x+c x^2\right )^{7/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac{2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}$

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(9*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^9) + (4*(a + b*x + c*x^2)^(7/2))/(63*(b^2 - 4*a*
c)^2*d^10*(b + 2*c*x)^7)

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Rubi [A]  time = 0.0336197, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.077, Rules used = {693, 682} $\frac{4 \left (a+b x+c x^2\right )^{7/2}}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^7}+\frac{2 \left (a+b x+c x^2\right )^{7/2}}{9 d^{10} \left (b^2-4 a c\right ) (b+2 c x)^9}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x]

[Out]

(2*(a + b*x + c*x^2)^(7/2))/(9*(b^2 - 4*a*c)*d^10*(b + 2*c*x)^9) + (4*(a + b*x + c*x^2)^(7/2))/(63*(b^2 - 4*a*
c)^2*d^10*(b + 2*c*x)^7)

Rule 693

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-2*b*d*(d + e*x)^(m
+ 1)*(a + b*x + c*x^2)^(p + 1))/(d^2*(m + 1)*(b^2 - 4*a*c)), x] + Dist[(b^2*(m + 2*p + 3))/(d^2*(m + 1)*(b^2
- 4*a*c)), Int[(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] && LtQ[m, -1] && (IntegerQ[2*p] || (IntegerQ[m] && Rationa
lQ[p]) || IntegerQ[(m + 2*p + 3)/2])

Rule 682

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

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{10}} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac{2 \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^8} \, dx}{9 \left (b^2-4 a c\right ) d^2}\\ &=\frac{2 \left (a+b x+c x^2\right )^{7/2}}{9 \left (b^2-4 a c\right ) d^{10} (b+2 c x)^9}+\frac{4 \left (a+b x+c x^2\right )^{7/2}}{63 \left (b^2-4 a c\right )^2 d^{10} (b+2 c x)^7}\\ \end{align*}

Mathematica [A]  time = 0.0550576, size = 62, normalized size = 0.78 $\frac{2 (a+x (b+c x))^{7/2} \left (4 c \left (2 c x^2-7 a\right )+9 b^2+8 b c x\right )}{63 d^{10} \left (b^2-4 a c\right )^2 (b+2 c x)^9}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(b*d + 2*c*d*x)^10,x]

[Out]

(2*(a + x*(b + c*x))^(7/2)*(9*b^2 + 8*b*c*x + 4*c*(-7*a + 2*c*x^2)))/(63*(b^2 - 4*a*c)^2*d^10*(b + 2*c*x)^9)

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Maple [A]  time = 0.042, size = 70, normalized size = 0.9 \begin{align*} -{\frac{-16\,{c}^{2}{x}^{2}-16\,bcx+56\,ac-18\,{b}^{2}}{63\, \left ( 2\,cx+b \right ) ^{9}{d}^{10} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,x)

[Out]

-2/63*(-8*c^2*x^2-8*b*c*x+28*a*c-9*b^2)*(c*x^2+b*x+a)^(7/2)/(2*c*x+b)^9/d^10/(16*a^2*c^2-8*a*b^2*c+b^4)

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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((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**(5/2)/(2*c*d*x+b*d)**10,x)

[Out]

Timed out

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Giac [B]  time = 4.57814, size = 2496, normalized size = 31.59 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^10,x, algorithm="giac")

[Out]

1/2016*(4032*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^14*c^(15/2) + 28224*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^13*b*
c^7 + 90048*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*b^2*c^(13/2) + 6720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^12*
a*c^(15/2) + 173376*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^11*b^3*c^6 + 40320*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))
^11*a*b*c^7 + 225792*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*b^4*c^(11/2) + 100800*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^10*a*b^2*c^(13/2) + 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^10*a^2*c^(15/2) + 212352*(sqrt(c)*x - sq
rt(c*x^2 + b*x + a))^9*b^5*c^5 + 134400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^3*c^6 + 100800*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^9*a^2*b*c^7 + 151200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*b^6*c^(9/2) + 96768*(sqrt(c)
*x - sqrt(c*x^2 + b*x + a))^8*a*b^4*c^(11/2) + 217728*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^2*b^2*c^(13/2) +
12096*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*c^(15/2) + 84672*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^7*c^
4 + 24192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^5*c^5 + 266112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b
^3*c^6 + 48384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b*c^7 + 38304*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*b
^8*c^(7/2) - 20160*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a*b^6*c^(9/2) + 205632*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^6*a^2*b^4*c^(11/2) + 72576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^3*b^2*c^(13/2) + 12096*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^6*a^4*c^(15/2) + 14112*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^9*c^3 - 24192*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^5*a*b^7*c^4 + 108864*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^5*c^5 + 48384*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^5*a^3*b^3*c^6 + 36288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^4*b*c^7 + 4176*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^10*c^(5/2) - 12960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^8*c^(7/2) +
43200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*b^6*c^(9/2) + 8640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^3*b
^4*c^(11/2) + 43200*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^4*b^2*c^(13/2) + 1728*(sqrt(c)*x - sqrt(c*x^2 + b*
x + a))^4*a^5*c^(15/2) + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^11*c^2 - 4416*(sqrt(c)*x - sqrt(c*x^2 + b
*x + a))^3*a*b^9*c^3 + 13824*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^2*b^7*c^4 - 6912*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))^3*a^3*b^5*c^5 + 25920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a^4*b^3*c^6 + 3456*(sqrt(c)*x - sqrt(c
*x^2 + b*x + a))^3*a^5*b*c^7 + 162*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*b^12*c^(3/2) - 1008*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))^2*a*b^10*c^(5/2) + 3456*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^2*b^8*c^(7/2) - 4608*(sqrt(
c)*x - sqrt(c*x^2 + b*x + a))^2*a^3*b^6*c^(9/2) + 8640*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^4*b^4*c^(11/2)
+ 1728*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^5*b^2*c^(13/2) + 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a^6*
c^(15/2) + 18*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^13*c - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^11*c^2
+ 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*b^9*c^3 - 1152*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^3*b^7*c^4 +
1728*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^4*b^5*c^5 + 576*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^6*b*c^7 + b^
14*sqrt(c) - 10*a*b^12*c^(3/2) + 48*a^2*b^10*c^(5/2) - 128*a^3*b^8*c^(7/2) + 224*a^4*b^6*c^(9/2) - 192*a^5*b^4
*c^(11/2) + 256*a^6*b^2*c^(13/2) - 64*a^7*c^(15/2))/((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c)^9*c^4*d^10)