\(\int (c+d (a+b x))^{5/2} \, dx\) [37]

Optimal result

Integrand size = 13, antiderivative size = 23 \[ \int (c+d (a+b x))^{5/2} \, dx=\frac {2 (c+d (a+b x))^{7/2}}{7 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \[ \int (c+d (a+b x))^{5/2} \, dx=\frac {2 (d (a+b x)+c)^{7/2}}{7 b d} \]

[In]

Int[(c + d*(a + b*x))^(5/2),x]

[Out]

(2*(c + d*(a + b*x))^(7/2))/(7*b*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 33

Int[((a_.) + (b_.)*(u_))^(m_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a + b*x)^m, x], x, u], x]
/; FreeQ[{a, b, m}, x] && LinearQ[u, x] && NeQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (c+d x)^{5/2} \, dx,x,a+b x\right )}{b} \\ & = \frac {2 (c+d (a+b x))^{7/2}}{7 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (c+d (a+b x))^{5/2} \, dx=\frac {2 (c+a d+b d x)^{7/2}}{7 b d} \]

[In]

Integrate[(c + d*(a + b*x))^(5/2),x]

[Out]

(2*(c + a*d + b*d*x)^(7/2))/(7*b*d)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

method	result	size
gosper	\(\frac {2 \left (b d x +a d +c \right )^{\frac {7}{2}}}{7 d b}\)	\(20\)
derivativedivides	\(\frac {2 \left (b d x +a d +c \right )^{\frac {7}{2}}}{7 d b}\)	\(20\)
default	\(\frac {2 \left (b d x +a d +c \right )^{\frac {7}{2}}}{7 d b}\)	\(20\)
pseudoelliptic	\(\frac {2 \left (c +d \left (b x +a \right )\right )^{\frac {7}{2}}}{7 b d}\)	\(20\)
trager	\(\frac {2 \left (b^{3} d^{3} x^{3}+3 a \,b^{2} d^{3} x^{2}+3 a^{2} b \,d^{3} x +3 b^{2} c \,d^{2} x^{2}+a^{3} d^{3}+6 a b c \,d^{2} x +3 a^{2} c \,d^{2}+3 b \,c^{2} d x +3 a \,c^{2} d +c^{3}\right ) \sqrt {b d x +a d +c}}{7 b d}\)	\(108\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (19) = 38\).

Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int (c+d (a+b x))^{5/2} \, dx=\frac {2 \, {\left (b^{3} d^{3} x^{3} + a^{3} d^{3} + 3 \, a^{2} c d^{2} + 3 \, a c^{2} d + c^{3} + 3 \, {\left (a b^{2} d^{3} + b^{2} c d^{2}\right )} x^{2} + 3 \, {\left (a^{2} b d^{3} + 2 \, a b c d^{2} + b c^{2} d\right )} x\right )} \sqrt {b d x + a d + c}}{7 \, b d} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (17) = 34\).

Time = 1.44 (sec) , antiderivative size = 270, normalized size of antiderivative = 11.74 \[ \int (c+d (a+b x))^{5/2} \, dx=\begin {cases} c^{\frac {5}{2}} x & \text {for}\: b = 0 \wedge d = 0 \\x \left (a d + c\right )^{\frac {5}{2}} & \text {for}\: b = 0 \\c^{\frac {5}{2}} x & \text {for}\: d = 0 \\\frac {2 a^{3} d^{2} \sqrt {a d + b d x + c}}{7 b} + \frac {6 a^{2} d^{2} x \sqrt {a d + b d x + c}}{7} + \frac {6 a^{2} c d \sqrt {a d + b d x + c}}{7 b} + \frac {6 a b d^{2} x^{2} \sqrt {a d + b d x + c}}{7} + \frac {12 a c d x \sqrt {a d + b d x + c}}{7} + \frac {6 a c^{2} \sqrt {a d + b d x + c}}{7 b} + \frac {2 b^{2} d^{2} x^{3} \sqrt {a d + b d x + c}}{7} + \frac {6 b c d x^{2} \sqrt {a d + b d x + c}}{7} + \frac {6 c^{2} x \sqrt {a d + b d x + c}}{7} + \frac {2 c^{3} \sqrt {a d + b d x + c}}{7 b d} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((c**(5/2)*x, Eq(b, 0) & Eq(d, 0)), (x*(a*d + c)**(5/2), Eq(b, 0)), (c**(5/2)*x, Eq(d, 0)), (2*a**3*d
**2*sqrt(a*d + b*d*x + c)/(7*b) + 6*a**2*d**2*x*sqrt(a*d + b*d*x + c)/7 + 6*a**2*c*d*sqrt(a*d + b*d*x + c)/(7*
b) + 6*a*b*d**2*x**2*sqrt(a*d + b*d*x + c)/7 + 12*a*c*d*x*sqrt(a*d + b*d*x + c)/7 + 6*a*c**2*sqrt(a*d + b*d*x
+ c)/(7*b) + 2*b**2*d**2*x**3*sqrt(a*d + b*d*x + c)/7 + 6*b*c*d*x**2*sqrt(a*d + b*d*x + c)/7 + 6*c**2*x*sqrt(a
*d + b*d*x + c)/7 + 2*c**3*sqrt(a*d + b*d*x + c)/(7*b*d), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int (c+d (a+b x))^{5/2} \, dx=\frac {2 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {7}{2}}}{7 \, b d} \]

[In]

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

[Out]

2/7*((b*x + a)*d + c)^(7/2)/(b*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (19) = 38\).

Time = 0.29 (sec) , antiderivative size = 444, normalized size of antiderivative = 19.30 \[ \int (c+d (a+b x))^{5/2} \, dx=\frac {2 \, {\left (35 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} a^{2} d^{2} - 35 \, {\left (3 \, \sqrt {b d x + a d + c} a d - {\left (b d x + a d + c\right )}^{\frac {3}{2}} + 3 \, \sqrt {b d x + a d + c} c\right )} a^{2} d^{2} - 21 \, {\left (b d x + a d + c\right )}^{\frac {5}{2}} a d + 70 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} a c d - 70 \, {\left (3 \, \sqrt {b d x + a d + c} a d - {\left (b d x + a d + c\right )}^{\frac {3}{2}} + 3 \, \sqrt {b d x + a d + c} c\right )} a c d + 5 \, {\left (b d x + a d + c\right )}^{\frac {7}{2}} - 21 \, {\left (b d x + a d + c\right )}^{\frac {5}{2}} c + 35 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} c^{2} - 35 \, {\left (3 \, \sqrt {b d x + a d + c} a d - {\left (b d x + a d + c\right )}^{\frac {3}{2}} + 3 \, \sqrt {b d x + a d + c} c\right )} c^{2} + 7 \, {\left (15 \, \sqrt {b d x + a d + c} a^{2} d^{2} - 10 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} a d + 30 \, \sqrt {b d x + a d + c} a c d + 3 \, {\left (b d x + a d + c\right )}^{\frac {5}{2}} - 10 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {b d x + a d + c} c^{2}\right )} a d + 7 \, {\left (15 \, \sqrt {b d x + a d + c} a^{2} d^{2} - 10 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} a d + 30 \, \sqrt {b d x + a d + c} a c d + 3 \, {\left (b d x + a d + c\right )}^{\frac {5}{2}} - 10 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {b d x + a d + c} c^{2}\right )} c\right )}}{35 \, b d} \]

[In]

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

[Out]

2/35*(35*(b*d*x + a*d + c)^(3/2)*a^2*d^2 - 35*(3*sqrt(b*d*x + a*d + c)*a*d - (b*d*x + a*d + c)^(3/2) + 3*sqrt(
b*d*x + a*d + c)*c)*a^2*d^2 - 21*(b*d*x + a*d + c)^(5/2)*a*d + 70*(b*d*x + a*d + c)^(3/2)*a*c*d - 70*(3*sqrt(b
*d*x + a*d + c)*a*d - (b*d*x + a*d + c)^(3/2) + 3*sqrt(b*d*x + a*d + c)*c)*a*c*d + 5*(b*d*x + a*d + c)^(7/2) -
21*(b*d*x + a*d + c)^(5/2)*c + 35*(b*d*x + a*d + c)^(3/2)*c^2 - 35*(3*sqrt(b*d*x + a*d + c)*a*d - (b*d*x + a*
d + c)^(3/2) + 3*sqrt(b*d*x + a*d + c)*c)*c^2 + 7*(15*sqrt(b*d*x + a*d + c)*a^2*d^2 - 10*(b*d*x + a*d + c)^(3/
2)*a*d + 30*sqrt(b*d*x + a*d + c)*a*c*d + 3*(b*d*x + a*d + c)^(5/2) - 10*(b*d*x + a*d + c)^(3/2)*c + 15*sqrt(b
*d*x + a*d + c)*c^2)*a*d + 7*(15*sqrt(b*d*x + a*d + c)*a^2*d^2 - 10*(b*d*x + a*d + c)^(3/2)*a*d + 30*sqrt(b*d*
x + a*d + c)*a*c*d + 3*(b*d*x + a*d + c)^(5/2) - 10*(b*d*x + a*d + c)^(3/2)*c + 15*sqrt(b*d*x + a*d + c)*c^2)*
c)/(b*d)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 4.04 \[ \int (c+d (a+b x))^{5/2} \, dx=\frac {6\,x\,\sqrt {c+d\,\left (a+b\,x\right )}\,{\left (c+a\,d\right )}^2}{7}+\frac {2\,\sqrt {c+d\,\left (a+b\,x\right )}\,{\left (c+a\,d\right )}^3}{7\,b\,d}+\frac {2\,b^2\,d^2\,x^3\,\sqrt {c+d\,\left (a+b\,x\right )}}{7}+\frac {6\,b\,d\,x^2\,\sqrt {c+d\,\left (a+b\,x\right )}\,\left (c+a\,d\right )}{7} \]

[In]

int((c + d*(a + b*x))^(5/2),x)

[Out]

(6*x*(c + d*(a + b*x))^(1/2)*(c + a*d)^2)/7 + (2*(c + d*(a + b*x))^(1/2)*(c + a*d)^3)/(7*b*d) + (2*b^2*d^2*x^3
*(c + d*(a + b*x))^(1/2))/7 + (6*b*d*x^2*(c + d*(a + b*x))^(1/2)*(c + a*d))/7