Optimal. Leaf size=257 \[ \frac {5 \sqrt {b} (b d-a e) (-3 a B e-4 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 e^{9/2}}-\frac {5 b \sqrt {a+b x} \sqrt {d+e x} (-3 a B e-4 A b e+7 b B d)}{4 e^4}+\frac {5 b (a+b x)^{3/2} \sqrt {d+e x} (-3 a B e-4 A b e+7 b B d)}{6 e^3 (b d-a e)}-\frac {2 (a+b x)^{5/2} (-3 a B e-4 A b e+7 b B d)}{3 e^2 \sqrt {d+e x} (b d-a e)}-\frac {2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
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Rubi [A] time = 0.20, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {78, 47, 50, 63, 217, 206} \[ -\frac {2 (a+b x)^{5/2} (-3 a B e-4 A b e+7 b B d)}{3 e^2 \sqrt {d+e x} (b d-a e)}+\frac {5 b (a+b x)^{3/2} \sqrt {d+e x} (-3 a B e-4 A b e+7 b B d)}{6 e^3 (b d-a e)}-\frac {5 b \sqrt {a+b x} \sqrt {d+e x} (-3 a B e-4 A b e+7 b B d)}{4 e^4}+\frac {5 \sqrt {b} (b d-a e) (-3 a B e-4 A b e+7 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 e^{9/2}}-\frac {2 (a+b x)^{7/2} (B d-A e)}{3 e (d+e x)^{3/2} (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{(d+e x)^{5/2}} \, dx &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{3 e (b d-a e) (d+e x)^{3/2}}+\frac {(7 b B d-4 A b e-3 a B e) \int \frac {(a+b x)^{5/2}}{(d+e x)^{3/2}} \, dx}{3 e (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 (7 b B d-4 A b e-3 a B e) (a+b x)^{5/2}}{3 e^2 (b d-a e) \sqrt {d+e x}}+\frac {(5 b (7 b B d-4 A b e-3 a B e)) \int \frac {(a+b x)^{3/2}}{\sqrt {d+e x}} \, dx}{3 e^2 (b d-a e)}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 (7 b B d-4 A b e-3 a B e) (a+b x)^{5/2}}{3 e^2 (b d-a e) \sqrt {d+e x}}+\frac {5 b (7 b B d-4 A b e-3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{6 e^3 (b d-a e)}-\frac {(5 b (7 b B d-4 A b e-3 a B e)) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{4 e^3}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 (7 b B d-4 A b e-3 a B e) (a+b x)^{5/2}}{3 e^2 (b d-a e) \sqrt {d+e x}}-\frac {5 b (7 b B d-4 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^4}+\frac {5 b (7 b B d-4 A b e-3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{6 e^3 (b d-a e)}+\frac {(5 b (b d-a e) (7 b B d-4 A b e-3 a B e)) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{8 e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 (7 b B d-4 A b e-3 a B e) (a+b x)^{5/2}}{3 e^2 (b d-a e) \sqrt {d+e x}}-\frac {5 b (7 b B d-4 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^4}+\frac {5 b (7 b B d-4 A b e-3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{6 e^3 (b d-a e)}+\frac {(5 (b d-a e) (7 b B d-4 A b e-3 a B e)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{4 e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 (7 b B d-4 A b e-3 a B e) (a+b x)^{5/2}}{3 e^2 (b d-a e) \sqrt {d+e x}}-\frac {5 b (7 b B d-4 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^4}+\frac {5 b (7 b B d-4 A b e-3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{6 e^3 (b d-a e)}+\frac {(5 (b d-a e) (7 b B d-4 A b e-3 a B e)) \operatorname {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{4 e^4}\\ &=-\frac {2 (B d-A e) (a+b x)^{7/2}}{3 e (b d-a e) (d+e x)^{3/2}}-\frac {2 (7 b B d-4 A b e-3 a B e) (a+b x)^{5/2}}{3 e^2 (b d-a e) \sqrt {d+e x}}-\frac {5 b (7 b B d-4 A b e-3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 e^4}+\frac {5 b (7 b B d-4 A b e-3 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{6 e^3 (b d-a e)}+\frac {5 \sqrt {b} (b d-a e) (7 b B d-4 A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 e^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.20, size = 113, normalized size = 0.44 \[ \frac {2 (a+b x)^{7/2} \left (-\frac {\left (\frac {b (d+e x)}{b d-a e}\right )^{3/2} (-3 a B e-4 A b e+7 b B d) \, _2F_1\left (\frac {3}{2},\frac {7}{2};\frac {9}{2};\frac {e (a+b x)}{a e-b d}\right )}{b}-7 A e+7 B d\right )}{21 e (d+e x)^{3/2} (a e-b d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 5.81, size = 855, normalized size = 3.33 \[ \left [\frac {15 \, {\left (7 \, B b^{2} d^{4} - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3} e + {\left (3 \, B a^{2} + 4 \, A a b\right )} d^{2} e^{2} + {\left (7 \, B b^{2} d^{2} e^{2} - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d e^{3} + {\left (3 \, B a^{2} + 4 \, A a b\right )} e^{4}\right )} x^{2} + 2 \, {\left (7 \, B b^{2} d^{3} e - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} e^{2} + {\left (3 \, B a^{2} + 4 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt {\frac {b}{e}} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 4 \, {\left (2 \, b e^{2} x + b d e + a e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {\frac {b}{e}} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (6 \, B b^{2} e^{3} x^{3} - 105 \, B b^{2} d^{3} - 8 \, A a^{2} e^{3} + 5 \, {\left (23 \, B a b + 12 \, A b^{2}\right )} d^{2} e - 8 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d e^{2} - 3 \, {\left (7 \, B b^{2} d e^{2} - {\left (9 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x^{2} - 2 \, {\left (70 \, B b^{2} d^{2} e - {\left (79 \, B a b + 40 \, A b^{2}\right )} d e^{2} + 4 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{48 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}}, -\frac {15 \, {\left (7 \, B b^{2} d^{4} - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{3} e + {\left (3 \, B a^{2} + 4 \, A a b\right )} d^{2} e^{2} + {\left (7 \, B b^{2} d^{2} e^{2} - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d e^{3} + {\left (3 \, B a^{2} + 4 \, A a b\right )} e^{4}\right )} x^{2} + 2 \, {\left (7 \, B b^{2} d^{3} e - 2 \, {\left (5 \, B a b + 2 \, A b^{2}\right )} d^{2} e^{2} + {\left (3 \, B a^{2} + 4 \, A a b\right )} d e^{3}\right )} x\right )} \sqrt {-\frac {b}{e}} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {b x + a} \sqrt {e x + d} \sqrt {-\frac {b}{e}}}{2 \, {\left (b^{2} e x^{2} + a b d + {\left (b^{2} d + a b e\right )} x\right )}}\right ) - 2 \, {\left (6 \, B b^{2} e^{3} x^{3} - 105 \, B b^{2} d^{3} - 8 \, A a^{2} e^{3} + 5 \, {\left (23 \, B a b + 12 \, A b^{2}\right )} d^{2} e - 8 \, {\left (2 \, B a^{2} + 5 \, A a b\right )} d e^{2} - 3 \, {\left (7 \, B b^{2} d e^{2} - {\left (9 \, B a b + 4 \, A b^{2}\right )} e^{3}\right )} x^{2} - 2 \, {\left (70 \, B b^{2} d^{2} e - {\left (79 \, B a b + 40 \, A b^{2}\right )} d e^{2} + 4 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} e^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {e x + d}}{24 \, {\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.41, size = 534, normalized size = 2.08 \[ -\frac {5 \, {\left (7 \, B b^{2} d^{2} {\left | b \right |} - 10 \, B a b d {\left | b \right |} e - 4 \, A b^{2} d {\left | b \right |} e + 3 \, B a^{2} {\left | b \right |} e^{2} + 4 \, A a b {\left | b \right |} e^{2}\right )} e^{\left (-\frac {9}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{4 \, \sqrt {b}} + \frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (B b^{5} d {\left | b \right |} e^{6} - B a b^{4} {\left | b \right |} e^{7}\right )} {\left (b x + a\right )}}{b^{4} d e^{7} - a b^{3} e^{8}} - \frac {7 \, B b^{6} d^{2} {\left | b \right |} e^{5} - 10 \, B a b^{5} d {\left | b \right |} e^{6} - 4 \, A b^{6} d {\left | b \right |} e^{6} + 3 \, B a^{2} b^{4} {\left | b \right |} e^{7} + 4 \, A a b^{5} {\left | b \right |} e^{7}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} - \frac {20 \, {\left (7 \, B b^{7} d^{3} {\left | b \right |} e^{4} - 17 \, B a b^{6} d^{2} {\left | b \right |} e^{5} - 4 \, A b^{7} d^{2} {\left | b \right |} e^{5} + 13 \, B a^{2} b^{5} d {\left | b \right |} e^{6} + 8 \, A a b^{6} d {\left | b \right |} e^{6} - 3 \, B a^{3} b^{4} {\left | b \right |} e^{7} - 4 \, A a^{2} b^{5} {\left | b \right |} e^{7}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (7 \, B b^{8} d^{4} {\left | b \right |} e^{3} - 24 \, B a b^{7} d^{3} {\left | b \right |} e^{4} - 4 \, A b^{8} d^{3} {\left | b \right |} e^{4} + 30 \, B a^{2} b^{6} d^{2} {\left | b \right |} e^{5} + 12 \, A a b^{7} d^{2} {\left | b \right |} e^{5} - 16 \, B a^{3} b^{5} d {\left | b \right |} e^{6} - 12 \, A a^{2} b^{6} d {\left | b \right |} e^{6} + 3 \, B a^{4} b^{4} {\left | b \right |} e^{7} + 4 \, A a^{3} b^{5} {\left | b \right |} e^{7}\right )}}{b^{4} d e^{7} - a b^{3} e^{8}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} d + {\left (b x + a\right )} b e - a b e\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 1250, normalized size = 4.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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