Optimal. Leaf size=339 \[ \frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (10 A c e (2 c d-b e)+B \left (8 b^2 e^2-13 b c d e+3 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 c^{5/2} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (5 A c e-4 b B e+3 B c d) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 c^{5/2} e \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{15 c^2}+\frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c} \]
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Rubi [A] time = 0.48, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {832, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (10 A c e (2 c d-b e)+B \left (8 b^2 e^2-13 b c d e+3 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 c^{5/2} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{15 c^2}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (5 A c e-4 b B e+3 B c d) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 c^{5/2} e \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 832
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx &=\frac {2 B (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 c}+\frac {2 \int \frac {\sqrt {d+e x} \left (-\frac {1}{2} (b B-5 A c) d+\frac {1}{2} (3 B c d-4 b B e+5 A c e) x\right )}{\sqrt {b x+c x^2}} \, dx}{5 c}\\ &=\frac {2 (3 B c d-4 b B e+5 A c e) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 B (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 c}+\frac {4 \int \frac {-\frac {1}{4} d \left (6 b B c d-15 A c^2 d-4 b^2 B e+5 A b c e\right )+\frac {1}{4} \left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c d e+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 c^2}\\ &=\frac {2 (3 B c d-4 b B e+5 A c e) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 B (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 c}-\frac {(d (c d-b e) (3 B c d-4 b B e+5 A c e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 c^2 e}+\frac {\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c d e+8 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{15 c^2 e}\\ &=\frac {2 (3 B c d-4 b B e+5 A c e) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 B (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 c}-\frac {\left (d (c d-b e) (3 B c d-4 b B e+5 A c e) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{15 c^2 e \sqrt {b x+c x^2}}+\frac {\left (\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c d e+8 b^2 e^2\right )\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{15 c^2 e \sqrt {b x+c x^2}}\\ &=\frac {2 (3 B c d-4 b B e+5 A c e) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 B (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 c}+\frac {\left (\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c d e+8 b^2 e^2\right )\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}{15 c^2 e \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) (3 B c d-4 b B e+5 A c 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}{15 c^2 e \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=\frac {2 (3 B c d-4 b B e+5 A c e) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 B (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 c}+\frac {2 \sqrt {-b} \left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c d e+8 b^2 e^2\right )\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 )}{15 c^{5/2} e \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) (3 B c d-4 b B e+5 A c 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 )}{15 c^{5/2} e \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 1.84, size = 356, normalized size = 1.05 \[ \frac {2 \sqrt {x} \left (\frac {(b+c x) (d+e x) \left (10 A c e (2 c d-b e)+B \left (8 b^2 e^2-13 b c d e+3 c^2 d^2\right )\right )}{c e \sqrt {x}}+i x \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (10 A c e (2 c d-b e)+B \left (8 b^2 e^2-13 b c d e+3 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-\frac {i x \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (b e-c d) \left (-b c (10 A e+9 B d)+15 A c^2 d+8 b^2 B e\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )}{b}+\sqrt {x} (b+c x) (d+e x) (5 A c e+B (-4 b e+6 c d+3 c e x))\right )}{15 c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e x^{2} + A d + {\left (B d + A e\right )} x\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 1144, normalized size = 3.37 \[ \frac {2 \sqrt {e x +d}\, \sqrt {\left (c x +b \right ) x}\, \left (3 B \,c^{4} e^{3} x^{4}+5 A \,c^{4} e^{3} x^{3}-B b \,c^{3} e^{3} x^{3}+9 B \,c^{4} d \,e^{2} x^{3}+5 A b \,c^{3} e^{3} x^{2}+5 A \,c^{4} d \,e^{2} x^{2}-4 B \,b^{2} c^{2} e^{3} x^{2}+5 B b \,c^{3} d \,e^{2} x^{2}+6 B \,c^{4} d^{2} e \,x^{2}+10 \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^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-30 \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^{2} \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}}\, A \,b^{2} c^{2} d \,e^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+20 \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} 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}}\, A b \,c^{3} d^{2} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+5 A b \,c^{3} d \,e^{2} x -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^{4} e^{3} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+21 \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^{2} \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-4 \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^{2} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-16 \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} e \EllipticE \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+7 \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} e \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )-4 B \,b^{2} c^{2} d \,e^{2} x +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 \,c^{3} d^{3} \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 \,c^{3} d^{3} \EllipticF \left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right )+6 B b \,c^{3} d^{2} e x \right )}{15 \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{4} e 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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {c\,x^2+b\,x}} \,d x \]
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 \[ \int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\sqrt {x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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