Optimal. Leaf size=413 \[ \frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (5 A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2-152 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 \sqrt {c} e^5 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (40 A c e (2 c d-b e)-B \left (3 b^2 e^2-88 b c d e+128 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 \sqrt {c} e^5 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {b x+c x^2} (e x (-10 A c e-3 b B e+16 B c d)-5 A e (8 c d-3 b e)+4 B d (16 c d-9 b e))}{15 e^4 \sqrt {d+e x}}+\frac {2 \left (b x+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}} \]
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Rubi [A] time = 0.50, antiderivative size = 413, 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 = {812, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (5 A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2-152 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 \sqrt {c} e^5 \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (40 A c e (2 c d-b e)-B \left (3 b^2 e^2-88 b c d e+128 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 \sqrt {c} e^5 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}+\frac {2 \left (b x+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}}-\frac {2 \sqrt {b x+c x^2} (e x (-10 A c e-3 b B e+16 B c d)-5 A e (8 c d-3 b e)+4 B d (16 c d-9 b e))}{15 e^4 \sqrt {d+e x}} \]
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) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx &=\frac {2 (8 B d-5 A e+3 B e x) \left (b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {2 \int \frac {\left (\frac {1}{2} b (8 B d-5 A e)+\frac {1}{2} (16 B c d-3 b B e-10 A c e) x\right ) \sqrt {b x+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^2}\\ &=-\frac {2 (4 B d (16 c d-9 b e)-5 A e (8 c d-3 b e)+e (16 B c d-3 b B e-10 A c e) x) \sqrt {b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}+\frac {4 \int \frac {\frac {1}{4} b (4 B d (16 c d-9 b e)-5 A e (8 c d-3 b e))-\frac {1}{4} \left (40 A c e (2 c d-b e)-B \left (128 c^2 d^2-88 b c d e+3 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 e^4}\\ &=-\frac {2 (4 B d (16 c d-9 b e)-5 A e (8 c d-3 b e)+e (16 B c d-3 b B e-10 A c e) x) \sqrt {b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {\left (40 A c e (2 c d-b e)-B \left (128 c^2 d^2-88 b c d e+3 b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{15 e^5}+\frac {\left (5 A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (128 c^2 d^2-152 b c d e+39 b^2 e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 e^5}\\ &=-\frac {2 (4 B d (16 c d-9 b e)-5 A e (8 c d-3 b e)+e (16 B c d-3 b B e-10 A c e) x) \sqrt {b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {\left (\left (40 A c e (2 c d-b e)-B \left (128 c^2 d^2-88 b c d e+3 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 e^5 \sqrt {b x+c x^2}}+\frac {\left (\left (5 A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (128 c^2 d^2-152 b c d e+39 b^2 e^2\right )\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{15 e^5 \sqrt {b x+c x^2}}\\ &=-\frac {2 (4 B d (16 c d-9 b e)-5 A e (8 c d-3 b e)+e (16 B c d-3 b B e-10 A c e) x) \sqrt {b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {\left (\left (40 A c e (2 c d-b e)-B \left (128 c^2 d^2-88 b c d e+3 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 e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (\left (5 A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (128 c^2 d^2-152 b c d e+39 b^2 e^2\right )\right ) \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 e^5 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (4 B d (16 c d-9 b e)-5 A e (8 c d-3 b e)+e (16 B c d-3 b B e-10 A c e) x) \sqrt {b x+c x^2}}{15 e^4 \sqrt {d+e x}}+\frac {2 (8 B d-5 A e+3 B e x) \left (b x+c x^2\right )^{3/2}}{15 e^2 (d+e x)^{3/2}}-\frac {2 \sqrt {-b} \left (40 A c e (2 c d-b e)-B \left (128 c^2 d^2-88 b c d e+3 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 \sqrt {c} e^5 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {-b} \left (5 A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (128 c^2 d^2-152 b c d e+39 b^2 e^2\right )\right ) \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 \sqrt {c} e^5 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 2.13, size = 436, normalized size = 1.06 \[ \frac {2 (x (b+c x))^{3/2} \left (i e x^{3/2} \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (5 A c e (8 c d-5 b e)+B \left (-3 b^2 e^2+52 b c d e-64 c^2 d^2\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i e x^{3/2} \sqrt {\frac {b}{c}} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (40 A c e (2 c d-b e)+B \left (-3 b^2 e^2+88 b c d e-128 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\frac {(b+c x) (d+e x) \left (40 A c e (b e-2 c d)+B \left (3 b^2 e^2-88 b c d e+128 c^2 d^2\right )\right )}{c}+\frac {e x (b+c x) \left (5 A e \left (c \left (8 d^2+10 d e x+e^2 x^2\right )-b e (3 d+4 e x)\right )+B \left (b e \left (36 d^2+47 d e x+6 e^2 x^2\right )-c \left (64 d^3+80 d^2 e x+8 d e^2 x^2-3 e^3 x^3\right )\right )\right )}{d+e x}\right )}{15 e^5 x^2 (b+c x)^2 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B c x^{3} + A b x + {\left (B b + A c\right )} x^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]
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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 2367, normalized size = 5.73 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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