### 3.1219 $$\int (b d+2 c d x)^3 (a+b x+c x^2)^{5/2} \, dx$$

Optimal. Leaf size=59 $\frac{4}{63} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}$

[Out]

(4*(b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^(7/2))/63 + (2*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^(7/2))/9

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Rubi [A]  time = 0.0264323, antiderivative size = 59, 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 = {692, 629} $\frac{4}{63} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*(b^2 - 4*a*c)*d^3*(a + b*x + c*x^2)^(7/2))/63 + (2*d^3*(b + 2*c*x)^2*(a + b*x + c*x^2)^(7/2))/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)^3 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{1}{9} \left (2 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx\\ &=\frac{4}{63} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0614551, size = 44, normalized size = 0.75 $\frac{2}{63} d^3 (a+x (b+c x))^{7/2} \left (4 c \left (7 c x^2-2 a\right )+9 b^2+28 b c x\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*d^3*(a + x*(b + c*x))^(7/2)*(9*b^2 + 28*b*c*x + 4*c*(-2*a + 7*c*x^2)))/63

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Maple [A]  time = 0.044, size = 41, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -56\,{c}^{2}{x}^{2}-56\,bcx+16\,ac-18\,{b}^{2} \right ){d}^{3}}{63} \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)^3*(c*x^2+b*x+a)^(5/2),x)

[Out]

-2/63*(c*x^2+b*x+a)^(7/2)*(-28*c^2*x^2-28*b*c*x+8*a*c-9*b^2)*d^3

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.11147, size = 467, normalized size = 7.92 \begin{align*} \frac{2}{63} \,{\left (28 \, c^{5} d^{3} x^{8} + 112 \, b c^{4} d^{3} x^{7} +{\left (177 \, b^{2} c^{3} + 76 \, a c^{4}\right )} d^{3} x^{6} +{\left (139 \, b^{3} c^{2} + 228 \, a b c^{3}\right )} d^{3} x^{5} + 5 \,{\left (11 \, b^{4} c + 51 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{3} x^{4} +{\left (9 \, b^{5} + 130 \, a b^{3} c + 120 \, a^{2} b c^{2}\right )} d^{3} x^{3} +{\left (27 \, a b^{4} + 87 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{3} x^{2} +{\left (27 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{3} x +{\left (9 \, a^{3} b^{2} - 8 \, a^{4} c\right )} d^{3}\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)^3*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

2/63*(28*c^5*d^3*x^8 + 112*b*c^4*d^3*x^7 + (177*b^2*c^3 + 76*a*c^4)*d^3*x^6 + (139*b^3*c^2 + 228*a*b*c^3)*d^3*
x^5 + 5*(11*b^4*c + 51*a*b^2*c^2 + 12*a^2*c^3)*d^3*x^4 + (9*b^5 + 130*a*b^3*c + 120*a^2*b*c^2)*d^3*x^3 + (27*a
*b^4 + 87*a^2*b^2*c + 4*a^3*c^2)*d^3*x^2 + (27*a^2*b^3 + 4*a^3*b*c)*d^3*x + (9*a^3*b^2 - 8*a^4*c)*d^3)*sqrt(c*
x^2 + b*x + a)

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Sympy [B]  time = 7.86453, size = 559, normalized size = 9.47 \begin{align*} - \frac{16 a^{4} c d^{3} \sqrt{a + b x + c x^{2}}}{63} + \frac{2 a^{3} b^{2} d^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{8 a^{3} b c d^{3} x \sqrt{a + b x + c x^{2}}}{63} + \frac{8 a^{3} c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{63} + \frac{6 a^{2} b^{3} d^{3} x \sqrt{a + b x + c x^{2}}}{7} + \frac{58 a^{2} b^{2} c d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{21} + \frac{80 a^{2} b c^{2} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{21} + \frac{40 a^{2} c^{3} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{21} + \frac{6 a b^{4} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{260 a b^{3} c d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{63} + \frac{170 a b^{2} c^{2} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{21} + \frac{152 a b c^{3} d^{3} x^{5} \sqrt{a + b x + c x^{2}}}{21} + \frac{152 a c^{4} d^{3} x^{6} \sqrt{a + b x + c x^{2}}}{63} + \frac{2 b^{5} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{110 b^{4} c d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{63} + \frac{278 b^{3} c^{2} d^{3} x^{5} \sqrt{a + b x + c x^{2}}}{63} + \frac{118 b^{2} c^{3} d^{3} x^{6} \sqrt{a + b x + c x^{2}}}{21} + \frac{32 b c^{4} d^{3} x^{7} \sqrt{a + b x + c x^{2}}}{9} + \frac{8 c^{5} d^{3} 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)**3*(c*x**2+b*x+a)**(5/2),x)

[Out]

-16*a**4*c*d**3*sqrt(a + b*x + c*x**2)/63 + 2*a**3*b**2*d**3*sqrt(a + b*x + c*x**2)/7 + 8*a**3*b*c*d**3*x*sqrt
(a + b*x + c*x**2)/63 + 8*a**3*c**2*d**3*x**2*sqrt(a + b*x + c*x**2)/63 + 6*a**2*b**3*d**3*x*sqrt(a + b*x + c*
x**2)/7 + 58*a**2*b**2*c*d**3*x**2*sqrt(a + b*x + c*x**2)/21 + 80*a**2*b*c**2*d**3*x**3*sqrt(a + b*x + c*x**2)
/21 + 40*a**2*c**3*d**3*x**4*sqrt(a + b*x + c*x**2)/21 + 6*a*b**4*d**3*x**2*sqrt(a + b*x + c*x**2)/7 + 260*a*b
**3*c*d**3*x**3*sqrt(a + b*x + c*x**2)/63 + 170*a*b**2*c**2*d**3*x**4*sqrt(a + b*x + c*x**2)/21 + 152*a*b*c**3
*d**3*x**5*sqrt(a + b*x + c*x**2)/21 + 152*a*c**4*d**3*x**6*sqrt(a + b*x + c*x**2)/63 + 2*b**5*d**3*x**3*sqrt(
a + b*x + c*x**2)/7 + 110*b**4*c*d**3*x**4*sqrt(a + b*x + c*x**2)/63 + 278*b**3*c**2*d**3*x**5*sqrt(a + b*x +
c*x**2)/63 + 118*b**2*c**3*d**3*x**6*sqrt(a + b*x + c*x**2)/21 + 32*b*c**4*d**3*x**7*sqrt(a + b*x + c*x**2)/9
+ 8*c**5*d**3*x**8*sqrt(a + b*x + c*x**2)/9

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Giac [B]  time = 1.17485, size = 389, normalized size = 6.59 \begin{align*} \frac{2}{63} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left ({\left ({\left ({\left ({\left (28 \,{\left (c^{5} d^{3} x + 4 \, b c^{4} d^{3}\right )} x + \frac{177 \, b^{2} c^{11} d^{3} + 76 \, a c^{12} d^{3}}{c^{8}}\right )} x + \frac{139 \, b^{3} c^{10} d^{3} + 228 \, a b c^{11} d^{3}}{c^{8}}\right )} x + \frac{5 \,{\left (11 \, b^{4} c^{9} d^{3} + 51 \, a b^{2} c^{10} d^{3} + 12 \, a^{2} c^{11} d^{3}\right )}}{c^{8}}\right )} x + \frac{9 \, b^{5} c^{8} d^{3} + 130 \, a b^{3} c^{9} d^{3} + 120 \, a^{2} b c^{10} d^{3}}{c^{8}}\right )} x + \frac{27 \, a b^{4} c^{8} d^{3} + 87 \, a^{2} b^{2} c^{9} d^{3} + 4 \, a^{3} c^{10} d^{3}}{c^{8}}\right )} x + \frac{27 \, a^{2} b^{3} c^{8} d^{3} + 4 \, a^{3} b c^{9} d^{3}}{c^{8}}\right )} x + \frac{9 \, a^{3} b^{2} c^{8} d^{3} - 8 \, a^{4} c^{9} d^{3}}{c^{8}}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*sqrt(c*x^2 + b*x + a)*(((((((28*(c^5*d^3*x + 4*b*c^4*d^3)*x + (177*b^2*c^11*d^3 + 76*a*c^12*d^3)/c^8)*x +
(139*b^3*c^10*d^3 + 228*a*b*c^11*d^3)/c^8)*x + 5*(11*b^4*c^9*d^3 + 51*a*b^2*c^10*d^3 + 12*a^2*c^11*d^3)/c^8)*
x + (9*b^5*c^8*d^3 + 130*a*b^3*c^9*d^3 + 120*a^2*b*c^10*d^3)/c^8)*x + (27*a*b^4*c^8*d^3 + 87*a^2*b^2*c^9*d^3 +
4*a^3*c^10*d^3)/c^8)*x + (27*a^2*b^3*c^8*d^3 + 4*a^3*b*c^9*d^3)/c^8)*x + (9*a^3*b^2*c^8*d^3 - 8*a^4*c^9*d^3)/
c^8)