Optimal. Leaf size=283 \[ \frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (-6 A c e-b B e+8 B c d) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}} \]
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Rubi [A] time = 0.29, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {812, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {b x+c x^2} (-3 A e+4 B d+B e x)}{3 e^2 \sqrt {d+e x}}+\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (-6 A c e-b B e+8 B c d) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx &=\frac {2 (4 B d-3 A e+B e x) \sqrt {b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {2 \int \frac {\frac {1}{2} b (4 B d-3 A e)+\frac {1}{2} (8 B c d-b B e-6 A c e) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 e^2}\\ &=\frac {2 (4 B d-3 A e+B e x) \sqrt {b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {(8 B c d-b B e-6 A c e) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 e^3}+\frac {(B d (8 c d-5 b e)-3 A e (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 e^3}\\ &=\frac {2 (4 B d-3 A e+B e x) \sqrt {b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {\left ((8 B c d-b B e-6 A c e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 e^3 \sqrt {b x+c x^2}}+\frac {\left ((B d (8 c d-5 b e)-3 A e (2 c d-b e)) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 e^3 \sqrt {b x+c x^2}}\\ &=\frac {2 (4 B d-3 A e+B e x) \sqrt {b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {\left ((8 B c d-b B e-6 A c e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 e^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left ((B d (8 c d-5 b e)-3 A e (2 c d-b e)) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 e^3 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 (4 B d-3 A e+B e x) \sqrt {b x+c x^2}}{3 e^2 \sqrt {d+e x}}-\frac {2 \sqrt {-b} (8 B c d-b B e-6 A c e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} (B d (8 c d-5 b e)-3 A e (2 c d-b e)) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 \sqrt {c} e^3 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 1.26, size = 269, normalized size = 0.95 \[ \frac {2 \left (b e x (b+c x) (-3 A e+4 B d+B e x)+\sqrt {\frac {b}{c}} \left (i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (-3 A c e-b B e+4 B c d) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (-6 A c e-b B e+8 B c d) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) (6 A c e+b B e-8 B c d)\right )\right )}{3 b e^3 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.22, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 804, normalized size = 2.84 \[ \frac {2 \sqrt {\left (c x +b \right ) x}\, \sqrt {e x +d}\, \left (B \,c^{3} e^{2} x^{3}-3 A \,c^{3} e^{2} x^{2}+B b \,c^{2} e^{2} x^{2}+4 B \,c^{3} d e \,x^{2}-6 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A \,b^{2} c \,e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+3 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A \,b^{2} c \,e^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+6 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A b \,c^{2} d e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-6 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, A b \,c^{2} d e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-3 A b \,c^{2} e^{2} x -\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B \,b^{3} e^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+9 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B \,b^{2} c d e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-5 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B \,b^{2} c d e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-8 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B b \,c^{2} d^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+8 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {\left (e x +d \right ) c}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, B b \,c^{2} d^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+4 B b \,c^{2} d e x \right )}{3 \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{2} e^{3} x} \]
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 \[ \int \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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