Optimal. Leaf size=295 \[ -\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (b B-2 A c) (c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {d+e x}} \]
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Rubi [A] time = 0.31, antiderivative size = 295, 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 = {818, 843, 715, 112, 110, 117, 116} \[ -\frac {2 \sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (b B-2 A c) (c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{(-b)^{3/2} c^{3/2} \sqrt {b x+c x^2} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 818
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 \int \frac {\frac {1}{2} b (b B+A c) d e+\frac {1}{2} e \left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac {2 \sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {((b B-2 A c) d (c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{b^2 c}+\frac {\left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac {2 \sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {\left ((b B-2 A c) d (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}{b^2 c \sqrt {b x+c x^2}}+\frac {\left (\left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{b^2 c \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {\left (\left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \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}{b^2 c \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left ((b B-2 A c) d (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}{b^2 c \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 \sqrt {d+e x} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 \left (2 A c^2 d+2 b^2 B e-b c (B d+A e)\right ) \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 )}{(-b)^{3/2} c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 (b B-2 A c) d (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 )}{(-b)^{3/2} c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 1.67, size = 302, normalized size = 1.02 \[ \frac {2 \left (b (d+e x) (x (b B-A c) (c d-b e)-A c d (b+c x))+\sqrt {\frac {b}{c}} \left (i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (-b c (A e+B d)+2 A c^2 d+2 b^2 B e\right )-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (A c-2 b B) (c d-b e) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{b^3 c \sqrt {x (b+c x)} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e x^{2} + A d + {\left (B d + A e\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 898, normalized size = 3.04 \[ \frac {2 \left (A b \,c^{3} e^{2} x^{2}-2 A \,c^{4} d e \,x^{2}-B \,b^{2} c^{2} e^{2} x^{2}+B b \,c^{3} d e \,x^{2}+\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^{3} 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^{2} d e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+2 \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^{2} d e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+2 \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^{3} d^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-2 \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^{3} d^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-2 A \,c^{4} d^{2} x -2 \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^{4} 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}}\, B \,b^{3} c d e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-\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} c d e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-\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^{2} d^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+\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^{2} d^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-B \,b^{2} c^{2} d e x +B b \,c^{3} d^{2} x -A b \,c^{3} d^{2}\right ) \sqrt {\left (c x +b \right ) x}}{\left (c x +b \right ) \sqrt {e x +d}\, b^{2} c^{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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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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