Optimal. Leaf size=359 \[ -\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (c x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (c d-b e) (8 c d-b e)\right )}{3 b^4 d \sqrt {b x+c x^2} (c d-b e)}+\frac {16 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}} \]
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Rubi [A] time = 0.40, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {736, 822, 843, 715, 112, 110, 117, 116} \[ \frac {2 \sqrt {d+e x} \left (c x \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+b (c d-b e) (8 c d-b e)\right )}{3 b^4 d \sqrt {b x+c x^2} (c d-b e)}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {16 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (2 c d-b e) F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} \sqrt {b x+c x^2} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rule 116
Rule 117
Rule 715
Rule 736
Rule 822
Rule 843
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {-4 c d+\frac {b e}{2}-3 c e x}{\sqrt {d+e x} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {4 \int \frac {\frac {1}{4} b c d e (8 c d-7 b e)+\frac {1}{4} c e \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {(8 c (2 c d-b e)) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4}-\frac {\left (c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 d (c d-b e)}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}+\frac {\left (8 c (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 b^4 \sqrt {b x+c x^2}}-\frac {\left (c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {\left (c \left (16 c^2 d^2-16 b c d e+b^2 e^2\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}{3 b^4 d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (8 c (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 b^4 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ &=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 b^2 \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b (c d-b e) (8 c d-b e)+c \left (16 c^2 d^2-16 b c d e+b^2 e^2\right ) x\right )}{3 b^4 d (c d-b e) \sqrt {b x+c x^2}}-\frac {2 \sqrt {c} \left (16 c^2 d^2-16 b c d e+b^2 e^2\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 )}{3 (-b)^{7/2} d (c d-b e) \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {16 \sqrt {c} (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 (-b)^{7/2} \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [C] time = 0.91, size = 375, normalized size = 1.04 \[ \frac {2 \left (b (d+e x) \left (b c^2 d x^2 (c d-b e)+c^2 d x^2 (b+c x) (8 c d-7 b e)+x (b+c x)^2 (c d-b e) (8 c d-b e)+b d (b+c x)^2 (b e-c d)\right )-c x \sqrt {\frac {b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (b^2 e^2-9 b c d e+8 c^2 d^2\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i b e x^{3/2} \sqrt {\frac {b}{c x}+1} \sqrt {\frac {d}{e x}+1} \left (b^2 e^2-16 b c d e+16 c^2 d^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+\sqrt {\frac {b}{c}} (b+c x) (d+e x) \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )\right )\right )}{3 b^5 d (x (b+c x))^{3/2} \sqrt {d+e x} (c d-b e)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + b x} \sqrt {e x + d}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, b^{2} c x^{4} + b^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 1362, normalized size = 3.79 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]
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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