\(\int (b d+2 c d x) (a+b x+c x^2)^{5/2} \, dx\) [1221]

Optimal result

Integrand size = 24, antiderivative size = 19 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{7} d \left (a+b x+c x^2\right )^{7/2} \]

[Out]

2/7*d*(c*x^2+b*x+a)^(7/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {643} \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{7} d \left (a+b x+c x^2\right )^{7/2} \]

[In]

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

[Out]

(2*d*(a + b*x + c*x^2)^(7/2))/7

Rule 643

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*} \text {integral}& = \frac {2}{7} d \left (a+b x+c x^2\right )^{7/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{7} d (a+x (b+c x))^{7/2} \]

[In]

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

[Out]

(2*d*(a + x*(b + c*x))^(7/2))/7

Maple [A] (verified)

Time = 2.94 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
gosper	\(\frac {2 d \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7}\)	\(16\)
default	\(\frac {2 d \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7}\)	\(16\)
pseudoelliptic	\(\frac {2 d \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7}\)	\(16\)
risch	\(\frac {2 d \left (c^{3} x^{6}+3 b \,c^{2} x^{5}+3 a \,x^{4} c^{2}+3 b^{2} c \,x^{4}+6 a b c \,x^{3}+b^{3} x^{3}+3 a^{2} c \,x^{2}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{7}\)	\(94\)
trager	\(d \left (\frac {2}{7} c^{3} x^{6}+\frac {6}{7} b \,c^{2} x^{5}+\frac {6}{7} a \,x^{4} c^{2}+\frac {6}{7} b^{2} c \,x^{4}+\frac {12}{7} a b c \,x^{3}+\frac {2}{7} b^{3} x^{3}+\frac {6}{7} a^{2} c \,x^{2}+\frac {6}{7} a \,b^{2} x^{2}+\frac {6}{7} a^{2} b x +\frac {2}{7} a^{3}\right ) \sqrt {c \,x^{2}+b x +a}\)	\(97\)

[In]

int((2*c*d*x+b*d)*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/7*d*(c*x^2+b*x+a)^(7/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (15) = 30\).

Time = 0.65 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.95 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{7} \, {\left (c^{3} d x^{6} + 3 \, b c^{2} d x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} d x^{4} + 3 \, a^{2} b d x + {\left (b^{3} + 6 \, a b c\right )} d x^{3} + a^{3} d + 3 \, {\left (a b^{2} + a^{2} c\right )} d x^{2}\right )} \sqrt {c x^{2} + b x + a} \]

[In]

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

[Out]

2/7*(c^3*d*x^6 + 3*b*c^2*d*x^5 + 3*(b^2*c + a*c^2)*d*x^4 + 3*a^2*b*d*x + (b^3 + 6*a*b*c)*d*x^3 + a^3*d + 3*(a*
b^2 + a^2*c)*d*x^2)*sqrt(c*x^2 + b*x + a)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (17) = 34\).

Time = 0.23 (sec) , antiderivative size = 260, normalized size of antiderivative = 13.68 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2 a^{3} d \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a^{2} b d x \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a^{2} c d x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a b^{2} d x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {12 a b c d x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 a c^{2} d x^{4} \sqrt {a + b x + c x^{2}}}{7} + \frac {2 b^{3} d x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 b^{2} c d x^{4} \sqrt {a + b x + c x^{2}}}{7} + \frac {6 b c^{2} d x^{5} \sqrt {a + b x + c x^{2}}}{7} + \frac {2 c^{3} d x^{6} \sqrt {a + b x + c x^{2}}}{7} \]

[In]

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

[Out]

2*a**3*d*sqrt(a + b*x + c*x**2)/7 + 6*a**2*b*d*x*sqrt(a + b*x + c*x**2)/7 + 6*a**2*c*d*x**2*sqrt(a + b*x + c*x
**2)/7 + 6*a*b**2*d*x**2*sqrt(a + b*x + c*x**2)/7 + 12*a*b*c*d*x**3*sqrt(a + b*x + c*x**2)/7 + 6*a*c**2*d*x**4
*sqrt(a + b*x + c*x**2)/7 + 2*b**3*d*x**3*sqrt(a + b*x + c*x**2)/7 + 6*b**2*c*d*x**4*sqrt(a + b*x + c*x**2)/7
+ 6*b*c**2*d*x**5*sqrt(a + b*x + c*x**2)/7 + 2*c**3*d*x**6*sqrt(a + b*x + c*x**2)/7

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} d \]

[In]

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

[Out]

2/7*(c*x^2 + b*x + a)^(7/2)*d

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} d \]

[In]

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

[Out]

2/7*(c*x^2 + b*x + a)^(7/2)*d

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2\,d\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7} \]

[In]

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

[Out]

(2*d*(a + b*x + c*x^2)^(7/2))/7