Optimal. Leaf size=23 \[ \frac {2 (d (a+b x)+c)^{7/2}}{7 b d} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32} \begin {gather*} \frac {2 (d (a+b x)+c)^{7/2}}{7 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 32
Rule 33
Rubi steps
\begin {align*} \int (c+d (a+b x))^{5/2} \, dx &=\frac {\operatorname {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] time = 0.02, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (d (a+b x)+c)^{7/2}}{7 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (a d+b d x+c)^{7/2}}{7 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.48, size = 104, normalized size = 4.52 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.64, size = 444, normalized size = 19.30 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 20, normalized size = 0.87 \begin {gather*} \frac {2 \left (b d x +a d +c \right )^{\frac {7}{2}}}{7 b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {7}{2}}}{7 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.18, size = 93, normalized size = 4.04 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 71.80, size = 270, normalized size = 11.74 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________