Optimal. Leaf size=489 \[ -\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {105 e^3 (a+b x) \sqrt {b d-a e} (-11 a B e+3 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {105 e^3 (a+b x) \sqrt {d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rubi [A] time = 0.42, antiderivative size = 489, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {770, 78, 47, 50, 63, 208} \begin {gather*} -\frac {21 e^2 (d+e x)^{5/2} (-11 a B e+3 A b e+8 b B d)}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {35 e^3 (a+b x) (d+e x)^{3/2} (-11 a B e+3 A b e+8 b B d)}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}+\frac {105 e^3 (a+b x) \sqrt {d+e x} (-11 a B e+3 A b e+8 b B d)}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (a+b x) \sqrt {b d-a e} (-11 a B e+3 A b e+8 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{11/2} (A b-a B)}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {(d+e x)^{9/2} (-11 a B e+3 A b e+8 b B d)}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {3 e (d+e x)^{7/2} (-11 a B e+3 A b e+8 b B d)}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 208
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx &=\frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(A+B x) (d+e x)^{9/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (b^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{128 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (b^2 d-a b e\right ) (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^6 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (b^2 d-a b e\right )^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^2 \left (b^2 d-a b e\right )^2 (8 b B d+3 A b e-11 a B e) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{64 b^8 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {105 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) \sqrt {d+e x}}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {35 e^3 (8 b B d+3 A b e-11 a B e) (a+b x) (d+e x)^{3/2}}{64 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (8 b B d+3 A b e-11 a B e) (d+e x)^{5/2}}{64 b^4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (8 b B d+3 A b e-11 a B e) (d+e x)^{7/2}}{32 b^3 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(8 b B d+3 A b e-11 a B e) (d+e x)^{9/2}}{24 b^2 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(A b-a B) (d+e x)^{11/2}}{4 b (b d-a e) (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 \sqrt {b d-a e} (8 b B d+3 A b e-11 a B e) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 117, normalized size = 0.24 \begin {gather*} \frac {(d+e x)^{11/2} \left (\frac {e^3 (a+b x)^4 (-11 a B e+3 A b e+8 b B d) \, _2F_1\left (4,\frac {11}{2};\frac {13}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^4}+11 (a B-A b)\right )}{44 b (a+b x)^3 \sqrt {(a+b x)^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 69.12, size = 726, normalized size = 1.48 \begin {gather*} \frac {(-a e-b e x) \left (\frac {105 \left (11 a^2 B e^5-3 a A b e^5-19 a b B d e^4+3 A b^2 d e^4+8 b^2 B d^2 e^3\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{64 b^{13/2} \sqrt {a e-b d}}-\frac {e^3 \sqrt {d+e x} \left (-3465 a^5 B e^5+945 a^4 A b e^5-12705 a^4 b B e^4 (d+e x)+16380 a^4 b B d e^4+3465 a^3 A b^2 e^4 (d+e x)-3780 a^3 A b^2 d e^4-30870 a^3 b^2 B d^2 e^3-16863 a^3 b^2 B e^3 (d+e x)^2+47355 a^3 b^2 B d e^3 (d+e x)+5670 a^2 A b^3 d^2 e^3+4599 a^2 A b^3 e^3 (d+e x)^2-10395 a^2 A b^3 d e^3 (d+e x)+28980 a^2 b^3 B d^3 e^2-65835 a^2 b^3 B d^2 e^2 (d+e x)-9207 a^2 b^3 B e^2 (d+e x)^3+45990 a^2 b^3 B d e^2 (d+e x)^2-3780 a A b^4 d^3 e^2+10395 a A b^4 d^2 e^2 (d+e x)+2511 a A b^4 e^2 (d+e x)^3-9198 a A b^4 d e^2 (d+e x)^2-13545 a b^4 B d^4 e+40425 a b^4 B d^3 e (d+e x)-41391 a b^4 B d^2 e (d+e x)^2-1408 a b^4 B e (d+e x)^4+15903 a b^4 B d e (d+e x)^3+945 A b^5 d^4 e-3465 A b^5 d^3 e (d+e x)+4599 A b^5 d^2 e (d+e x)^2+384 A b^5 e (d+e x)^4-2511 A b^5 d e (d+e x)^3+2520 b^5 B d^5-9240 b^5 B d^4 (d+e x)+12264 b^5 B d^3 (d+e x)^2-6696 b^5 B d^2 (d+e x)^3+128 b^5 B (d+e x)^5+1024 b^5 B d (d+e x)^4\right )}{192 b^6 (a e+b (d+e x)-b d)^4}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 1414, normalized size = 2.89
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.52, size = 872, normalized size = 1.78 \begin {gather*} \frac {105 \, {\left (8 \, B b^{2} d^{2} e^{3} - 19 \, B a b d e^{4} + 3 \, A b^{2} d e^{4} + 11 \, B a^{2} e^{5} - 3 \, A a b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} - \frac {1320 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{3} - 3560 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{3} + 3224 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{3} - 984 \, \sqrt {x e + d} B b^{5} d^{5} e^{3} - 3615 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{4} + 975 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{4} + 12975 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{4} - 2295 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{4} - 14825 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{4} + 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{4} + 5481 \, \sqrt {x e + d} B a b^{4} d^{4} e^{4} - 561 \, \sqrt {x e + d} A b^{5} d^{4} e^{4} + 2295 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{5} - 975 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{5} - 15270 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{5} + 4590 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{5} + 25131 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{5} - 5787 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{5} - 12084 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{5} + 2244 \, \sqrt {x e + d} A a b^{4} d^{3} e^{5} + 5855 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{6} - 2295 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{6} - 18683 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{6} + 5787 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{6} + 13206 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{6} - 3366 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{6} + 5153 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{7} - 1929 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{7} - 7164 \, \sqrt {x e + d} B a^{4} b d e^{7} + 2244 \, \sqrt {x e + d} A a^{3} b^{2} d e^{7} + 1545 \, \sqrt {x e + d} B a^{5} e^{8} - 561 \, \sqrt {x e + d} A a^{4} b e^{8}}{192 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{4} b^{6} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{10} e^{3} + 12 \, \sqrt {x e + d} B b^{10} d e^{3} - 15 \, \sqrt {x e + d} B a b^{9} e^{4} + 3 \, \sqrt {x e + d} A b^{10} e^{4}\right )}}{3 \, b^{15} \mathrm {sgn}\left ({\left (x e + d\right )} b e - b d e + a e^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 2430, normalized size = 4.97
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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