Optimal. Leaf size=527 \[ -\frac {2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{3/2} \left (-5 b c (A e+B d)+10 A c^2 d+6 b^2 B e\right )}{5 b^2 c^2}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right )}{15 b^2 c^3}-\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (-8 b^3 c e^2 (5 A e+16 B d)+b^2 c^2 d e (95 A e+103 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}} \]
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Rubi [A] time = 0.91, antiderivative size = 527, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {818, 832, 843, 715, 112, 110, 117, 116} \[ \frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x} \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right )}{15 b^2 c^3}-\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}}+\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 c^2 d e (95 A e+103 B d)-8 b^3 c e^2 (5 A e+16 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt {b x+c x^2}}+\frac {2 e \sqrt {b x+c x^2} (d+e x)^{3/2} \left (-5 b c (A e+B d)+10 A c^2 d+6 b^2 B e\right )}{5 b^2 c^2} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 818
Rule 832
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (d+e x)^{5/2} \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 {(d+e x)^{3/2} \left (\frac {1}{2} b (b B+5 A c) d e+\frac {1}{2} e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{b^2 c}\\ &=-\frac {2 (d+e x)^{5/2} \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 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 b^2 c^2}+\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {1}{4} b d e \left (15 A c^2 d-6 b^2 B e+5 b c (2 B d+A e)\right )+\frac {1}{4} e \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) x\right )}{\sqrt {b x+c x^2}} \, dx}{5 b^2 c^2}\\ &=-\frac {2 (d+e x)^{5/2} \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 e \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 b^2 c^3}+\frac {2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 b^2 c^2}+\frac {8 \int \frac {\frac {1}{8} b d e \left (15 A c^3 d^2+24 b^3 B e^2+45 b c^2 d (B d+A e)-b^2 c e (61 B d+20 A e)\right )+\frac {1}{8} e \left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 b^2 c^3}\\ &=-\frac {2 (d+e x)^{5/2} \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 e \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 b^2 c^3}+\frac {2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 b^2 c^2}-\frac {\left (d (c d-b e) \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{15 b^2 c^3}+\frac {\left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{15 b^2 c^3}\\ &=-\frac {2 (d+e x)^{5/2} \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 e \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 b^2 c^3}+\frac {2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 b^2 c^2}-\frac {\left (d (c d-b e) \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{15 b^2 c^3 \sqrt {b x+c x^2}}+\frac {\left (\left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{15 b^2 c^3 \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{5/2} \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 e \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 b^2 c^3}+\frac {2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 b^2 c^2}+\frac {\left (\left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 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}{15 b^2 c^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {\left (d (c d-b e) \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\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 b^2 c^3 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (d+e x)^{5/2} \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 e \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\right ) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 b^2 c^3}+\frac {2 e \left (10 A c^2 d+6 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 b^2 c^2}+\frac {2 \left (30 A c^4 d^3+48 b^4 B e^3-15 b c^3 d^2 (B d+3 A e)-8 b^3 c e^2 (16 B d+5 A e)+b^2 c^2 d e (103 B d+95 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 )}{15 (-b)^{3/2} c^{7/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 d (c d-b e) \left (30 A c^3 d^2-24 b^3 B e^2-15 b c^2 d (B d+2 A e)+b^2 c e (43 B d+20 A e)\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 (-b)^{3/2} c^{7/2} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 4.06, size = 493, normalized size = 0.94 \[ \frac {2 \left (b (d+e x) \left (b^2 e^2 x (b+c x) (5 A c e-9 b B e+16 B c d)+15 x (b B-A c) (c d-b e)^3-15 A c^3 d^3 (b+c x)+3 b^2 B c e^3 x^2 (b+c x)\right )+\sqrt {\frac {b}{c}} \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} (c d-b e) \left (8 b^2 c e (5 A e+13 B d)-15 b c^2 d (5 A e+4 B d)+15 A c^3 d^2-48 b^3 B e^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (-8 b^3 c e^2 (5 A e+16 B d)+b^2 c^2 d e (95 A e+103 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (-8 b^3 c e^2 (5 A e+16 B d)+b^2 c^2 d e (95 A e+103 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right )\right )\right )}{15 b^3 c^3 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B e^{3} x^{4} + A d^{3} + {\left (3 \, B d e^{2} + A e^{3}\right )} x^{3} + 3 \, {\left (B d^{2} e + A d e^{2}\right )} x^{2} + {\left (B d^{3} + 3 \, A d^{2} e\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 1766, normalized size = 3.35 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]
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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