Optimal. Leaf size=51 \[ -\frac {2 a A}{5 x^{5/2}}-\frac {2 (A b+a B)}{3 x^{3/2}}-\frac {2 (b B+A c)}{\sqrt {x}}+2 B c \sqrt {x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {779}
\begin {gather*} -\frac {2 (a B+A b)}{3 x^{3/2}}-\frac {2 a A}{5 x^{5/2}}-\frac {2 (A c+b B)}{\sqrt {x}}+2 B c \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 779
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{x^{7/2}} \, dx &=\int \left (\frac {a A}{x^{7/2}}+\frac {A b+a B}{x^{5/2}}+\frac {b B+A c}{x^{3/2}}+\frac {B c}{\sqrt {x}}\right ) \, dx\\ &=-\frac {2 a A}{5 x^{5/2}}-\frac {2 (A b+a B)}{3 x^{3/2}}-\frac {2 (b B+A c)}{\sqrt {x}}+2 B c \sqrt {x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 42, normalized size = 0.82 \begin {gather*} -\frac {2 (a (3 A+5 B x)+5 x (3 B x (b-c x)+A (b+3 c x)))}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 40, normalized size = 0.78
method | result | size |
derivativedivides | \(-\frac {2 a A}{5 x^{\frac {5}{2}}}-\frac {2 \left (A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {2 \left (A c +B b \right )}{\sqrt {x}}+2 B c \sqrt {x}\) | \(40\) |
default | \(-\frac {2 a A}{5 x^{\frac {5}{2}}}-\frac {2 \left (A b +B a \right )}{3 x^{\frac {3}{2}}}-\frac {2 \left (A c +B b \right )}{\sqrt {x}}+2 B c \sqrt {x}\) | \(40\) |
gosper | \(-\frac {2 \left (-15 B c \,x^{3}+15 A c \,x^{2}+15 b B \,x^{2}+5 A b x +5 B a x +3 A a \right )}{15 x^{\frac {5}{2}}}\) | \(42\) |
trager | \(-\frac {2 \left (-15 B c \,x^{3}+15 A c \,x^{2}+15 b B \,x^{2}+5 A b x +5 B a x +3 A a \right )}{15 x^{\frac {5}{2}}}\) | \(42\) |
risch | \(-\frac {2 \left (-15 B c \,x^{3}+15 A c \,x^{2}+15 b B \,x^{2}+5 A b x +5 B a x +3 A a \right )}{15 x^{\frac {5}{2}}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 40, normalized size = 0.78 \begin {gather*} 2 \, B c \sqrt {x} - \frac {2 \, {\left (15 \, {\left (B b + A c\right )} x^{2} + 3 \, A a + 5 \, {\left (B a + A b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.65, size = 39, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (15 \, B c x^{3} - 15 \, {\left (B b + A c\right )} x^{2} - 3 \, A a - 5 \, {\left (B a + A b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.32, size = 65, normalized size = 1.27 \begin {gather*} - \frac {2 A a}{5 x^{\frac {5}{2}}} - \frac {2 A b}{3 x^{\frac {3}{2}}} - \frac {2 A c}{\sqrt {x}} - \frac {2 B a}{3 x^{\frac {3}{2}}} - \frac {2 B b}{\sqrt {x}} + 2 B c \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.91, size = 42, normalized size = 0.82 \begin {gather*} 2 \, B c \sqrt {x} - \frac {2 \, {\left (15 \, B b x^{2} + 15 \, A c x^{2} + 5 \, B a x + 5 \, A b x + 3 \, A a\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.30, size = 42, normalized size = 0.82 \begin {gather*} 2\,B\,c\,\sqrt {x}-\frac {\left (2\,A\,c+2\,B\,b\right )\,x^2+\left (\frac {2\,A\,b}{3}+\frac {2\,B\,a}{3}\right )\,x+\frac {2\,A\,a}{5}}{x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________