Optimal. Leaf size=179 \[ \frac {c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{5/2}}-\frac {c^2 \sqrt {b x+c x^2} (8 b B-3 A c)}{64 b^2 x^{3/2}}-\frac {c \sqrt {b x+c x^2} (8 b B-3 A c)}{32 b x^{5/2}}-\frac {\left (b x+c x^2\right )^{3/2} (8 b B-3 A c)}{24 b x^{9/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {792, 662, 672, 660, 207} \begin {gather*} -\frac {c^2 \sqrt {b x+c x^2} (8 b B-3 A c)}{64 b^2 x^{3/2}}+\frac {c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{5/2}}-\frac {c \sqrt {b x+c x^2} (8 b B-3 A c)}{32 b x^{5/2}}-\frac {\left (b x+c x^2\right )^{3/2} (8 b B-3 A c)}{24 b x^{9/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 660
Rule 662
Rule 672
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{13/2}} \, dx &=-\frac {A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}}+\frac {\left (-\frac {13}{2} (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{11/2}} \, dx}{4 b}\\ &=-\frac {(8 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{24 b x^{9/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}}+\frac {(c (8 b B-3 A c)) \int \frac {\sqrt {b x+c x^2}}{x^{7/2}} \, dx}{16 b}\\ &=-\frac {c (8 b B-3 A c) \sqrt {b x+c x^2}}{32 b x^{5/2}}-\frac {(8 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{24 b x^{9/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}}+\frac {\left (c^2 (8 b B-3 A c)\right ) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{64 b}\\ &=-\frac {c (8 b B-3 A c) \sqrt {b x+c x^2}}{32 b x^{5/2}}-\frac {c^2 (8 b B-3 A c) \sqrt {b x+c x^2}}{64 b^2 x^{3/2}}-\frac {(8 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{24 b x^{9/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}}-\frac {\left (c^3 (8 b B-3 A c)\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{128 b^2}\\ &=-\frac {c (8 b B-3 A c) \sqrt {b x+c x^2}}{32 b x^{5/2}}-\frac {c^2 (8 b B-3 A c) \sqrt {b x+c x^2}}{64 b^2 x^{3/2}}-\frac {(8 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{24 b x^{9/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}}-\frac {\left (c^3 (8 b B-3 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{64 b^2}\\ &=-\frac {c (8 b B-3 A c) \sqrt {b x+c x^2}}{32 b x^{5/2}}-\frac {c^2 (8 b B-3 A c) \sqrt {b x+c x^2}}{64 b^2 x^{3/2}}-\frac {(8 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{24 b x^{9/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{4 b x^{13/2}}+\frac {c^3 (8 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{64 b^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.03, size = 62, normalized size = 0.35 \begin {gather*} \frac {(x (b+c x))^{5/2} \left (c^3 x^4 (8 b B-3 A c) \, _2F_1\left (\frac {5}{2},4;\frac {7}{2};\frac {c x}{b}+1\right )-5 A b^4\right )}{20 b^5 x^{13/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.07, size = 135, normalized size = 0.75 \begin {gather*} \frac {\left (8 b B c^3-3 A c^4\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{64 b^{5/2}}+\frac {\sqrt {b x+c x^2} \left (-48 A b^3-72 A b^2 c x-6 A b c^2 x^2+9 A c^3 x^3-64 b^3 B x-112 b^2 B c x^2-24 b B c^2 x^3\right )}{192 b^2 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 288, normalized size = 1.61 \begin {gather*} \left [-\frac {3 \, {\left (8 \, B b c^{3} - 3 \, A c^{4}\right )} \sqrt {b} x^{5} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (48 \, A b^{4} + 3 \, {\left (8 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{3} + 2 \, {\left (56 \, B b^{3} c + 3 \, A b^{2} c^{2}\right )} x^{2} + 8 \, {\left (8 \, B b^{4} + 9 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{384 \, b^{3} x^{5}}, -\frac {3 \, {\left (8 \, B b c^{3} - 3 \, A c^{4}\right )} \sqrt {-b} x^{5} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (48 \, A b^{4} + 3 \, {\left (8 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{3} + 2 \, {\left (56 \, B b^{3} c + 3 \, A b^{2} c^{2}\right )} x^{2} + 8 \, {\left (8 \, B b^{4} + 9 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{192 \, b^{3} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.32, size = 176, normalized size = 0.98 \begin {gather*} -\frac {\frac {3 \, {\left (8 \, B b c^{4} - 3 \, A c^{5}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {24 \, {\left (c x + b\right )}^{\frac {7}{2}} B b c^{4} + 40 \, {\left (c x + b\right )}^{\frac {5}{2}} B b^{2} c^{4} - 88 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{3} c^{4} + 24 \, \sqrt {c x + b} B b^{4} c^{4} - 9 \, {\left (c x + b\right )}^{\frac {7}{2}} A c^{5} + 33 \, {\left (c x + b\right )}^{\frac {5}{2}} A b c^{5} + 33 \, {\left (c x + b\right )}^{\frac {3}{2}} A b^{2} c^{5} - 9 \, \sqrt {c x + b} A b^{3} c^{5}}{b^{2} c^{4} x^{4}}}{192 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 185, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {\left (c x +b \right ) x}\, \left (9 A \,c^{4} x^{4} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-24 B b \,c^{3} x^{4} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-9 \sqrt {c x +b}\, A \sqrt {b}\, c^{3} x^{3}+24 \sqrt {c x +b}\, B \,b^{\frac {3}{2}} c^{2} x^{3}+6 \sqrt {c x +b}\, A \,b^{\frac {3}{2}} c^{2} x^{2}+112 \sqrt {c x +b}\, B \,b^{\frac {5}{2}} c \,x^{2}+72 \sqrt {c x +b}\, A \,b^{\frac {5}{2}} c x +64 \sqrt {c x +b}\, B \,b^{\frac {7}{2}} x +48 \sqrt {c x +b}\, A \,b^{\frac {7}{2}}\right )}{192 \sqrt {c x +b}\, b^{\frac {5}{2}} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{x^{\frac {13}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{x^{13/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________