Optimal. Leaf size=66 \[ -\frac {2 \sqrt {c+d x}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {4 d \sqrt {c+d x}}{3 (b c-a d)^2 \sqrt {a+b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37}
\begin {gather*} \frac {4 d \sqrt {c+d x}}{3 \sqrt {a+b x} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{3 (a+b x)^{3/2} (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx &=-\frac {2 \sqrt {c+d x}}{3 (b c-a d) (a+b x)^{3/2}}-\frac {(2 d) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{3 (b c-a d)}\\ &=-\frac {2 \sqrt {c+d x}}{3 (b c-a d) (a+b x)^{3/2}}+\frac {4 d \sqrt {c+d x}}{3 (b c-a d)^2 \sqrt {a+b x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 46, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {c+d x} (-b c+3 a d+2 b d x)}{3 (b c-a d)^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 55, normalized size = 0.83
method | result | size |
gosper | \(\frac {2 \sqrt {d x +c}\, \left (2 b d x +3 a d -b c \right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
default | \(-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs.
\(2 (54) = 108\).
time = 0.34, size = 118, normalized size = 1.79 \begin {gather*} \frac {2 \, {\left (2 \, b d x - b c + 3 \, a d\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 121 vs.
\(2 (54) = 108\).
time = 0.01, size = 136, normalized size = 2.06 \begin {gather*} -\frac {32 b \sqrt {b d} b d \left (-3 \left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2}-a b d+b^{2} c\right )}{2\cdot 6 \left |b\right | \left (\left (\sqrt {-a b d+b^{2} c+b d \left (a+b x\right )}-\sqrt {b d} \sqrt {a+b x}\right )^{2}+a b d-b^{2} c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.89, size = 71, normalized size = 1.08 \begin {gather*} \frac {\left (\frac {4\,d\,x}{3\,{\left (a\,d-b\,c\right )}^2}+\frac {6\,a\,d-2\,b\,c}{3\,b\,{\left (a\,d-b\,c\right )}^2}\right )\,\sqrt {c+d\,x}}{x\,\sqrt {a+b\,x}+\frac {a\,\sqrt {a+b\,x}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________